1. Chapter 5
Forecasting
To accompany Quantitative Analysis for Management, Tenth Edition, by Render, Stair, and Hanna
Power Point slides created by Jeff Heyl
© 2009 Prentice-Hall, Inc.

2. Introduction
Managers are always trying to reduce uncertainty and make better estimates of what will happen in the future
This is the main purpose of forecasting
Some firms use subjective methods
Seat-of-the pants methods, intuition, experience
There are also several quantitative techniques
Moving averages, exponential smoothing, trend projections, least squares regression analysis

3. Introduction Eight steps to forecasting :
Determine the use of the forecast—what objective are we trying to obtain?
Select the items or quantities that are to be forecasted
Determine the time horizon of the forecast
Select the forecasting model or models
Gather the data needed to make the forecast
Validate the forecasting model
Make the forecast
Implement the results

4. Introduction
These steps are a systematic way of initiating, designing, and implementing a forecasting system
When used regularly over time, data is collected routinely and calculations performed automatically
There is seldom one superior forecasting system
Different organizations may use different techniques
Whatever tool works best for a firm is the one they should use

5. Forecasting Techniques
Forecasting Models
Forecasting Techniques
Time-Series Methods
Qualitative
Models
Causal
Methods
Delphi
Methods
Jury of Executive
Opinion
Sales Force
Composite
Consumer
Market Survey
Moving
Average
Exponential Smoothing
Trend
Projections
Decomposition
Regression Analysis
Multiple
Regression
Figure 5.1

6. Time-Series Models
Time-series models attempt to predict the future based on the past
Common time-series models are
Moving average
Exponential smoothing
Trend projections
Decomposition
Regression analysis is used in trend projections and one type of decomposition model

7. Causal Models
Causal models use variables or factors that might influence the quantity being forecasted
The objective is to build a model with the best statistical relationship between the variable being forecast and the independent variables
Regression analysis is the most common technique used in causal modeling

8. Qualitative Models
Qualitative models incorporate judgmental or subjective factors
Useful when subjective factors are thought to be important or when accurate quantitative data is difficult to obtain
Common qualitative techniques are
Delphi method
Jury of executive opinion
Sales force composite
Consumer market surveys

9. Qualitative Models
Delphi Method – an iterative group process where (possibly geographically dispersed) respondents provide input to decision makers
Jury of Executive Opinion – collects opinions of a small group of high-level managers, possibly using statistical models for analysis
Sales Force Composite – individual salespersons estimate the sales in their region and the data is compiled at a district or national level
Consumer Market Survey – input is solicited from customers or potential customers regarding their purchasing plans

10. Scatter Diagrams
Scatter diagrams are helpful when forecasting time-series data because they depict the relationship between variables.
Radios
Televisions
Compact Discs

11. Scatter Diagrams
Wacker Distributors wants to forecast sales for three different products
YEAR
TELEVISION SETS
RADIOS
COMPACT DISC PLAYERS 1
250
300
110 2
310
100 3
320
120 4
330
140 5
340
170 6
350
150 7
360
160 8
370
190 9
380
200 10
390
Table 5.1

12. Scatter Diagrams Sales appear to be constant over time Sales = 250
330 –
250 –
200 –
150 –
100 –
50 –
| | | | | | | | | |
Time (Years)
Annual Sales of Televisions
(a)
Sales appear to be constant over time
Sales = 250
A good estimate of sales in year 11 is 250 televisions 
Figure 5.2

13. Scatter Diagrams
420 –
400 –
380 –
360 –
340 –
320 –
300 –
280 –
| | | | | | | | | |
Time (Years)
Annual Sales of Radios
(b)
Sales appear to be increasing at a constant rate of 10 radios per year
Sales = (Year)
A reasonable estimate of sales in year 11 is 400 televisions 
Figure 5.2

14. Scatter Diagrams
This trend line may not be perfectly accurate because of variation from year to year
Sales appear to be increasing
A forecast would probably be a larger figure each year
200 –
180 –
160 –
140 –
120 –
100 –
| | | | | | | | | |
Time (Years)
Annual Sales of CD Players
(c) 
Figure 5.2

15. Measures of Forecast Accuracy
We compare forecasted values with actual values to see how well one model works or to compare models
Forecast error = Actual value – Forecast value
One measure of accuracy is the mean absolute deviation (MAD)

16. Measures of Forecast Accuracy
Using a naïve forecasting model
YEAR
ACTUAL SALES OF CD PLAYERS
FORECAST SALES
ABSOLUTE VALUE OF ERRORS (DEVIATION), (ACTUAL – FORECAST) 1
110 — 2
100
|100 – 110| = 10 3
120
|120 – 110| = 20 4
140
|140 – 120| = 20 5
170
|170 – 140| = 30 6
150
|150 – 170| = 20 7
160
|160 – 150| = 10 8
190
|190 – 160| = 30 9
200
|200 – 190| = 10 10
|190 – 200| = 10 11
Sum of |errors| = 160
MAD = 160/9 = 17.8
Table 5.2

17. Measures of Forecast Accuracy
Using a naïve forecasting model
YEAR
ACTUAL SALES OF CD PLAYERS
FORECAST SALES
ABSOLUTE VALUE OF ERRORS (DEVIATION), (ACTUAL – FORECAST) 1
110 — 2
100
|100 – 110| = 10 3
120
|120 – 110| = 20 4
140
|140 – 120| = 20 5
170
|170 – 140| = 30 6
150
|150 – 170| = 20 7
160
|160 – 150| = 10 8
190
|190 – 160| = 30 9
200
|200 – 190| = 10 10
|190 – 200| = 10 11
Sum of |errors| = 160
MAD = 160/9 = 17.8
Table 5.2

18. Measures of Forecast Accuracy
There are other popular measures of forecast accuracy
The mean squared error
The mean absolute percent error
And bias is the average error and tells whether the forecast tends to be too high or too low and by how much. Thus, it can be negative or positive.

19. Measures of Forecast Accuracy
Year
Actual CD Sales
Forecast Sales
|Actual -Forecast| 1
110 2
100 10 3
120 20 4
140 5
170 30 6
150 7
160 8
190 9
200 11
Sum of |errors|
MAD
17.8

20. Hospital Days Forecast Error Example
Month
Forecast
Actual
JAN
250
243
FEB
320
315
MAR
275
286
APR
260
256
MAY
241
JUN
298
JUL
300
292
AUG
325
333
SEP
326
OCT
350
378
NOV
365
382
DEC
380
396
Ms. Smith forecasted total hospital inpatient days last year. Now that the actual data are known, she is reevaluating her forecasting model.
Compute the MAD, MSE, and MAPE for her forecast.

21. Hospital Days Forecast Error Example
Actual
|error|
error2
|error/actual|
JAN
250
243 7 49
0.03
FEB
320
315 5 25
0.02
MAR
275
286 11
121
0.04
APR
260
256 4 16
MAY
241 9 81
JUN
298 23
529
0.08
JUL
300
292 8 64
AUG
325
333
SEP
326 6 36
OCT
350
378 28
784
0.07
NOV
365
382 17
289
DEC
380
396
AVERAGE
MAD=
11.83
MSE=
192.83
MAPE=
.0381*100 =
3.81

22. Time-Series Forecasting Models
A time series is a sequence of evenly spaced events (weekly, monthly, quarterly, etc.)
Time-series forecasts predict the future based solely of the past values of the variable
Other variables, no matter how potentially valuable, are ignored

23. Decomposition of a Time-Series
A time series typically has four components
Trend (T) is the gradual upward or downward movement of the data over time
Seasonality (S) is a pattern of demand fluctuations above or below trend line that repeats at regular intervals
Cycles (C) are patterns in annual data that occur every several years
Random variations (R) are “blips” in the data caused by chance and unusual situations

24. Decomposition of a Time-Series
Demand for Product or Service
| | | |
Year Year Year Year
Trend Component
Seasonal Peaks
Actual Demand Line
Average Demand over 4 Years
Figure 5.3

25. Decomposition of a Time-Series
There are two general forms of time-series models
The multiplicative model
Demand = T x S x C x R
The additive model
Demand = T + S + C + R
Models may be combinations of these two forms
Forecasters often assume errors are normally distributed with a mean of zero

26. Moving Averages
Moving averages can be used when demand is relatively steady over time
The next forecast is the average of the most recent n data values from the time series
The most recent period of data is added and the oldest is dropped
This methods tends to smooth out short-term irregularities in the data series

27. Moving Averages Mathematically where = forecast for time period t + 1
= actual value in time period t
n = number of periods to average

28. Wallace Garden Supply Example
Wallace Garden Supply wants to forecast demand for its Storage Shed
They have collected data for the past year
They are using a three-month moving average to forecast demand (n = 3)

29. Wallace Garden Supply Example
MONTH
ACTUAL SHED SALES
THREE-MONTH MOVING AVERAGE
January 10
February 12
March 13
April 16
May 19
June 23
July 26
August 30
September 28
October 18
November
December 14 —
( )/3 = 11.67
( )/3 = 13.67
( )/3 = 16.00
( )/3 = 19.33
( )/3 = 22.67
( )/3 = 26.33
( )/3 = 28.00
( )/3 = 25.33
( )/3 = 20.67
( )/3 = 16.00
Table 5.3

30. Weighted Moving Averages
Weighted moving averages use weights to put more emphasis on recent periods
Often used when a trend or other pattern is emerging
Mathematically
where
wi = weight for the ith observation

31. Weighted Moving Averages
Both simple and weighted averages are effective in smoothing out fluctuations in the demand pattern in order to provide stable estimates
Problems
Increasing the size of n smoothes out fluctuations better, but makes the method less sensitive to real changes in the data
Moving averages can not pick up trends very well – they will always stay within past levels and not predict a change to a higher or lower level

32. Wallace Garden Supply Example
Wallace Garden Supply decides to try a weighted moving average model to forecast demand for its Storage Shed
They decide on the following weighting scheme
WEIGHTS APPLIED
PERIOD 3
Last month 2
Two months ago 1
Three months ago 6
3 x Sales last month + 2 x Sales two months ago + 1 X Sales three months ago
Sum of the weights

33. Wallace Garden Supply Example
MONTH
ACTUAL SHED SALES
THREE-MONTH WEIGHTED MOVING AVERAGE
January 10
February 12
March 13
April 16
May 19
June 23
July 26
August 30
September 28
October 18
November
December 14 —
[(3 X 13) + (2 X 12) + (10)]/6 = 12.17
[(3 X 16) + (2 X 13) + (12)]/6 = 14.33
[(3 X 19) + (2 X 16) + (13)]/6 = 17.00
[(3 X 23) + (2 X 19) + (16)]/6 = 20.50
[(3 X 26) + (2 X 23) + (19)]/6 = 23.83
[(3 X 30) + (2 X 26) + (23)]/6 = 27.50
[(3 X 28) + (2 X 30) + (26)]/6 = 28.33
[(3 X 18) + (2 X 28) + (30)]/6 = 23.33
[(3 X 16) + (2 X 18) + (28)]/6 = 18.67
[(3 X 14) + (2 X 16) + (18)]/6 = 15.33
Table 5.4

34. Wallace Garden Supply Example
Program 5.1A

35. Wallace Garden Supply Example
Program 5.1B

36. Exponential Smoothing
Exponential smoothing is easy to use and requires little record keeping of data
It is a type of moving average
New forecast = Last period’s forecast
+ (Last period’s actual demand
– Last period’s forecast)
Where  is a weight (or smoothing constant) with a value between 0 and 1 inclusive
A larger  gives more importance to recent data while a smaller value gives more importance to past data

37. Exponential Smoothing
Mathematically
where
Ft+1 = new forecast (for time period t + 1)
Ft = pervious forecast (for time period t)
 = smoothing constant (0 ≤  ≤ 1)
Yt = pervious period’s actual demand
The idea is simple – the new estimate is the old estimate plus some fraction of the error in the last period

38. Exponential Smoothing Example
In January, February’s demand for a certain car model was predicted to be 142
Actual February demand was 153 autos
Using a smoothing constant of  = 0.20, what is the forecast for March?
New forecast (for March demand) = (153 – 142)
= or 144 autos
If actual demand in March was 136 autos, the April forecast would be
New forecast (for April demand) = (136 – 144.2)
= or 143 autos

39. Selecting the Smoothing Constant
Selecting the appropriate value for  is key to obtaining a good forecast
The objective is always to generate an accurate forecast
The general approach is to develop trial forecasts with different values of  and select the  that results in the lowest MAD

40. Port of Baltimore Example
Exponential smoothing forecast for two values of 
QUARTER
ACTUAL TONNAGE UNLOADED
FORECAST USING  =0.10
FORECAST USING  =0.50 1
180
175 2
168
= (180 – 175)
177.5 3
159
= (168 – )
172.75 4
= (159 – )
165.88 5
190
= (175 – )
170.44 6
205
= (190 – )
180.22 7
= (205 – )
192.61 8
182
= (180 – )
186.30 9 ?
= (182 – )
184.15
Table 5.5

41. Selecting the Best Value of 
QUARTER
ACTUAL TONNAGE UNLOADED
FORECAST WITH  = 0.10
ABSOLUTE
DEVIATIONS FOR  = 0.10
FORECAST WITH  = 0.50
DEVIATIONS FOR  = 0.50 1
180
175
5…..
5…. 2
168
175.5
7.5..
177.5
9.5.. 3
159
174.75
15.75
172.75
13.75 4
173.18
1.82
165.88
9.12 5
190
173.36
16.64
170.44
19.56 6
205
175.02
29.98
180.22
24.78 7
178.02
1.98
192.61
12.61 8
182
178.22
3.78
186.30
4.3..
Sum of absolute deviations
82.45
98.63
MAD =
Σ|deviations| =
10.31
12.33 n
Best choice
Table 5.6

42. Port of Baltimore Example
Program 5.2A

43. Port of Baltimore Example
Program 5.2B

44. PM Computer: Moving Average Example
PM Computer assembles customized personal computers from generic parts
The owners purchase generic computer parts in volume at a discount from a variety of sources whenever they see a good deal.
It is important that they develop a good forecast of demand for their computers so they can purchase component parts efficiently.

45. PM Computers: Data Compute a 2-month moving average
Period Month Actual Demand
1 Jan 37
2 Feb 40
3 Mar 41
4 Apr 37
5 May 45
6 June 50
7 July 43
8 Aug 47
9 Sept 56
Compute a 2-month moving average
Compute a 3-month weighted average using weights of 4,2,1 for the past three months of data
Compute an exponential smoothing forecast using  = 0.7, previous forecast of 40
Using MAD, what forecast is most accurate?

46. PM Computers: Moving Average Solution
MAD
Exponential smoothing resulted in the lowest MAD.

47. Exponential Smoothing with Trend Adjustment
Like all averaging techniques, exponential smoothing does not respond to trends
A more complex model can be used that adjusts for trends
The basic approach is to develop an exponential smoothing forecast then adjust it for the trend
Forecast including trend (FITt) = New forecast (Ft)
+ Trend correction (Tt)

48. Exponential Smoothing with Trend Adjustment
The equation for the trend correction uses a new smoothing constant 
Tt is computed by
where
Tt+1 = smoothed trend for period t + 1
Tt = smoothed trend for preceding period
 = trend smooth constant that we select
Ft+1 = simple exponential smoothed forecast for period t + 1
Ft = forecast for pervious period

49. Selecting a Smoothing Constant
As with exponential smoothing, a high value of  makes the forecast more responsive to changes in trend
A low value of  gives less weight to the recent trend and tends to smooth out the trend
Values are generally selected using a trial-and-error approach based on the value of the MAD for different values of 
Simple exponential smoothing is often referred to as first-order smoothing
Trend-adjusted smoothing is called second-order, double smoothing, or Holt’s method

50. Trend Projection
Trend projection fits a trend line to a series of historical data points
The line is projected into the future for medium- to long-range forecasts
Several trend equations can be developed based on exponential or quadratic models
The simplest is a linear model developed using regression analysis

51. Trend Projection
Trend projections are used to forecast time-series data that exhibit a linear trend.
A trend line is simply a linear regression equation in which the independent variable (X) is the time period
Least squares may be used to determine a trend projection for future forecasts.
Least squares determines the trend line forecast by minimizing the mean squared error between the trend line forecasts and the actual observed values.
The independent variable is the time period and the dependent variable is the actual observed value in the time series.

52. Trend Projection The mathematical form is where = predicted value
b0 = intercept
b1 = slope of the line
X = time period (i.e., X = 1, 2, 3, …, n)

53. Trend Projection * Value of Dependent Variable Time Dist7 Dist5 Dist6
Figure 5.4

54. Midwestern Manufacturing Company Example
Midwestern Manufacturing Company has experienced the following demand for it’s electrical generators over the period of 2001 – 2007
YEAR
ELECTRICAL GENERATORS SOLD
2001 74
2002 79
2003 80
2004 90
2005
105
2006
142
2007
122
Table 5.7

55. Midwestern Manufacturing Company Example
Notice code instead of actual years
Program 5.3A

56. Midwestern Manufacturing Company Example
r2 says model predicts about 80% of the variability in demand
Significance level for F-test indicates a definite relationship
Program 5.3B

57. Midwestern Manufacturing Company Example
The forecast equation is
To project demand for 2008, we use the coding system to define X = 8
(sales in 2008) = (8)
= , or 141 generators
Likewise for X = 9
(sales in 2009) = (9)
= , or 152 generators

58. Midwestern Manufacturing Company Example
Generator Demand
Year
160 –
150 –
140 –
130 –
120 –
110 –
100 –
90 –
80 –
70 –
60 –
50 –
| | | | | | | | |  
Trend Line
Actual Demand Line
Figure 5.5

59. Midwestern Manufacturing Company Example
Program 5.4A

60. Midwestern Manufacturing Company Example
Program 5.4B

61. Seasonal Variations
Recurring variations over time may indicate the need for seasonal adjustments in the trend line
A seasonal index indicates how a particular season compares with an average season
When no trend is present, the seasonal index can be found by dividing the average value for a particular season by the average of all the data

62. Seasonal Variations
Eichler Supplies sells telephone answering machines
Data has been collected for the past two years sales of one particular model
They want to create a forecast that includes seasonality

63. AVERAGE TWO- YEAR DEMAND AVERAGE SEASONAL INDEX
Seasonal Variations
MONTH
SALES DEMAND
AVERAGE TWO- YEAR DEMAND
MONTHLY DEMAND
AVERAGE SEASONAL INDEX
YEAR 1
YEAR 2
January 80
100 90 94
0.957
February 85 75
0.851
March
0.904
April
110
1.064
May
115
131
123
1.309
June
120
1.223
July
105
1.117
August
September 95
October
November
December
Total average demand = 1,128
Average monthly demand = = 94
1,128
12 months
Seasonal index =
Average two-year demand
Average monthly demand
Table 5.8

64. Seasonal Variations The calculations for the seasonal indices are Jan.
July
Feb.
Aug.
Mar.
Sept.
Apr.
Oct.
May
Nov.
June
Dec.

65. Regression with Trend and Seasonal Components
Multiple regression can be used to forecast both trend and seasonal components in a time series
One independent variable is time
Dummy independent variables are used to represent the seasons
The model is an additive decomposition model
where
X1 = time period
X2 = 1 if quarter 2, 0 otherwise
X3 = 1 if quarter 3, 0 otherwise
X4 = 1 if quarter 4, 0 otherwise

66. Regression with Trend and Seasonal Components
Program 5.6A

67. Regression with Trend and Seasonal Components
Program 5.6B (partial)

68. Regression with Trend and Seasonal Components
The resulting regression equation is
Using the model to forecast sales for the first two quarters of next year
These are different from the results obtained using the multiplicative decomposition method
Use MAD and MSE to determine the best model

69. Regression with Trend and Seasonal Components
American Airlines original spare parts inventory system used only time-series methods to forecast the demand for spare parts
This method was slow to responds to even moderate changes in aircraft utilization let alone major fleet expansions
They developed a PC-based system named RAPS which uses linear regression to establish a relationship between monthly part removals and various functions of monthly flying hours
The computation now takes only one hour instead of the days the old system needed
Using RAPS provided a one time savings of $7 million and a recurring annual savings of nearly $1 million

70. Monitoring and Controlling Forecasts
Tracking signals can be used to monitor the performance of a forecast
Tacking signals are computed using the following equation
where

71. Monitoring and Controlling Forecasts
Upper Control Limit
Lower Control Limit
0 MADs + –
Time
Signal Tripped
Tracking Signal
Acceptable Range
Figure 5.7

72. Monitoring and Controlling Forecasts
Positive tracking signals indicate demand is greater than forecast
Negative tracking signals indicate demand is less than forecast
Some variation is expected, but a good forecast will have about as much positive error as negative error
Problems are indicated when the signal trips either the upper or lower predetermined limits
This indicates there has been an unacceptable amount of variation
Limits should be reasonable and may vary from item to item

73. Regression with Trend and Seasonal Components
How do you decide on the upper and lower limits?
Too small a value will trip the signal too often and too large will cause a bad forecast
Plossl & Wight – use maximums of ±4 MADs for high volume stock items and ±8 MADs for lower volume items
One MAD is equivalent to approximately 0.8 standard deviation so that ±4 MADs =3.2 s.d.
For a forecast to be “in control”, 89% of the errors are expected to fall within ±2 MADs, 98% with ±3 MADs or 99.9% within ±4 MADs whenever the errors are approximately normally distributed

74. Kimball’s Bakery Example
Tracking signal for quarterly sales of croissants
TIME PERIOD
FORECAST DEMAND
ACTUAL DEMAND
ERROR
RSFE
|FORECAST | | ERROR |
CUMULATIVE ERROR
MAD
TRACKING SIGNAL 1
100 90
–10 10
10.0 –1 2 95 –5
–15 5 15
7.5 –2 3
115
+15 30 4
110 40
125 +5 55
11.0
+0.5 6
140
+30
+35 85
14.2
+2.5

75. Forecasting at Disney
The Disney chairman receives a daily report from his main theme parks that contains only two numbers – the forecast of yesterday’s attendance at the parks and the actual attendance
An error close to zero (using MAPE as the measure) is expected
The annual forecast of total volume conducted in 1999 for the year 2000 resulted in a MAPE of 0

76. Using The Computer to Forecast
Spreadsheets can be used by small and medium-sized forecasting problems
More advanced programs (SAS, SPSS, Minitab) handle time-series and causal models
May automatically select best model parameters
Dedicated forecasting packages may be fully automatic
May be integrated with inventory planning and control