Forecasting
A forecast is a prediction of what will occur in the future
Example : forecast the weather sportscasters predict the winners of football games and managers of business firms attempt to predict how much of their product will be demanded in the future Trying to predict the future, making decisions in the present that will ensure the continued success of their firms. Often a manager will use judgment, opinion, or past experiences to forecast what will occur in the future
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Managers are always trying to reduce uncertainty and make better estimates of what will happen in the future
Forecasting Components A variety of forecasting methods are available for use depending on the time frame of the forecast and the existence of patterns.
Time Frames: Short-range (one to two months) Medium-range (two months to one or two years) Long-range (more than one or two years) Patterns: Trend Random variations Cycles Seasonal pattern
Forecasting Components
Trend - A long-term movement of the item being forecast.
Random variations - movements that are not predictable and follow no pattern.
Cycle - A movement, up or down, that repeats itself over a lengthy time span.
Seasonal pattern - Oscillating movement in demand that occurs periodically in the short run and is repetitive.
Forecasting Components
Forms of Forecast Movement: (a) Trend, (b) Cycle, (c) Seasonal Pattern, (d) Trend with Seasonal Pattern
Forecasting Methods
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Times Series - Statistical techniques that use historical data to predict future behavior.
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Regression Methods - Regression (or causal ) methods that attempt to develop a mathematical relationship between the item being forecast and factors that cause it to behave the way it does.
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Qualitative Methods - Methods using judgment, expertise and opinion to make forecasts.
Time Series Methods
Statistical techniques that make use of historical data collected over a long period of time.
Methods assume that what has occurred in the past will continue to occur in the future.
Forecasts based on only one factor - time.
Naïve method Ft+1 =At YEAR
ACTUAL SALES OF CD PLAYERS
FORECAST SALES
1
110
—
2
100
110
3
120
100
4
140
120
5
170
140
6
150
170
7
160
150
8
190
160
9
200
190
10
190
200
11
—
190
Time Series Methods Moving Average
Moving average uses values from the recent past to develop forecasts.
Useful for forecasting relatively stable items that do not display any trend or seasonal pattern.
Formula for: n Di Yt Yt 1 ... Yt n1 i 1 Ft 1 MA n n n where : n number of periods in the moving average D data in period i i
Time Series Methods Moving Average
Three- and Five-Month Moving Averages
Time Series Methods Moving Average
Three- and Five-Month Moving Averages
Time Series Methods Moving Average
Longer-period moving averages react more slowly to changes in demand than do shorter-period moving averages.
The appropriate number of periods to use often requires trial-and-error experimentation.
Moving average does not react well to changes (trends, seasonal effects, etc.) but is easy to use and inexpensive.
Good for short-term forecasting.
Time Series Methods Weighted Moving Average
In a weighted moving average, weights are assigned to the most recent data. Formula: n WMAn Wi Di i1 where Wi the weight for period i, between 0% and 100% Wi 1.00 Example : Paper clip company we ights 50% for October, 33% for September, 17% for August : 3 WMA Wi Di (.50)(90) (.33)(110) (.17)(130) 103.4 orders 3 i1
Time Series Methods Weighted Moving Average
Usually decreasing as you go back in time for example :
Ft = .5 At +.3 At-1 + .2 At-2
F9 = .5 A8 +.3 A7 + .2 A6
Example 2
Example3
Time Series Methods Exponential Smoothing (1 of 11)
Two forms: simple exponential smoothing and adjusted exponential smoothing.
Simple exponential smoothing: Ft + 1 = Dt + (1 - )Ft
where: Ft + 1 = the forecast for the next period Dt = actual demand in the present period Ft = the previously determined forecast for the present period = a weighting factor (smoothing constant). Example : F3= F2+ (A2-F2)
Time Series Methods Exponential Smoothing (2 of 11)
The most commonly used values of are between 0.10 and 0.50.
Default : F1=A1 = 0.3
Time Series Methods Exponential Smoothing (3 of 11) Example: PM Computer Services (see Table 15.4).
Exponential smoothing forecasts using smoothing constant of .30.
Forecast for period 2 (February): F2 = D1 + (1- )F1 = (.30)(.37) + (.70)(.37) = 37 units
Forecast for period 3 (March): F3 = D2 + (1- )F2 = (.30)(.40) + (.70)(37) = 37.9 units OR : F3= F2+ (A2-F2)
Time Series Methods Exponential Smoothing (4 of 11)
Table 15.4 Exponential Smoothing Forecasts, ď Ą = .30 and ď Ą = .50
Time Series Methods Exponential Smoothing (6 of 11)
Figure 15.3 Exponential Smoothing Forecasts
Time Series Methods Exponential Smoothing (5 of 11) ď Ž
Low smoothing constants are appropriate for stable data without trend; higher constants appropriate for data with trends.
Ex 1
Ex 2
Ex 3
Time Series Methods Exponential Smoothing (7 of 11) Adjusted exponential smoothing: exponential smoothing with a trend adjustment factor added. Formula: AFt + 1 = Ft + 1 + Tt+1 where: T = an exponentially smoothed trend factor Tt + 1 + (Ft + 1 - Ft) + (1 - )Tt Tt = the last period trend factor = smoothing constant for trend ( a value between zero and one).
Time Series Methods Exponential Smoothing (10 of 11) Adjusted forecast is consistently higher than the simple exponentially smoothed forecast.
It is more reflective of the generally increasing trend of the data.
Time Series Methods Exponential Smoothing (11 of 11)
Figure 15.4 Adjusted Exponentially Smoothed Forecast
Ex 1
Ex 2
Time Series Methods Linear Trend (5 of 5)
Figure 15.5 Linear Trend Line
Example 1
Example 2
Example 3
Time Series Methods Seasonal Adjustments (1 of 4) A seasonal pattern is a repetitive up-and-down movement in demand.
Seasonal patterns can occur on a quarterly, monthly, weekly, or daily basis. A seasonally adjusted forecast can be developed by multiplying the normal forecast by a seasonal factor. A seasonal factor can be determined by dividing the actual demand for each seasonal period by total annual demand: Si =Di/D
Time Series Methods Seasonal Adjustments (2 of 4)
Seasonal factors lie between zero and one and represent the portion of total annual demand assigned to each season. Seasonal factors are multiplied by annual demand to provide adjusted forecasts for each period.
Time Series Methods Seasonal Adjustments (3 of 4) Example: Wishbone Farms
Demand for Turkeys at Wishbone Farms
S1 = D1/ D = 42.0/148.7 = 0.28 S2 = D2/ D = 29.5/148.7 = 0.20 S3 = D3/ D = 21.9/148.7 = 0.15 S4 = D4/ D = 55.3/148.7 = 0.37
Time Series Methods Seasonal Adjustments (4 of 4) Multiply forecasted demand for entire year by seasonal factors to determine quarterly demand. Forecast for entire year (trend line for data): y = 40.97 + 4.30x = 40.97 + 4.30(4) = 58.17 Seasonally adjusted forecasts: SF1 = (S1)(F5) = (.28)(58.17) = 16.28 SF2 = (S2)(F5) = (.20)(58.17) = 11.63
SF3 = (S3)(F5) = (.15)(58.17) = 8.73 SF4 = (S4)(F5) = (.37)(58.17) = 21.53
Forecast Accuracy Overview Forecasts will always deviate from actual values. Difference between forecasts and actual values referred to as forecast error.
Would like forecast error to be as small as possible. If error is large, either technique being used is the wrong one, or parameters need adjusting. Measures of forecast errors: Mean Absolute deviation (MAD) Mean absolute percentage deviation (MAPD) Cumulative error (E bar) Average error, or bias (E)
Forecast Accuracy Mean Absolute Deviation (1 of 7) MAD is the average absolute difference between the forecast and actual demand.
Formula: Dt Ft MAD n where: t the period number Dt demand in period t Ft the forecast for period t n the total number of periods
Forecast Accuracy Mean Absolute Deviation (2 of 7) Example: PM Computer Services (see Table). Compare accuracies of different forecasts using MAD:
Dt Ft 53.41 MAD 4.85 n 11
Forecast Accuracy Mean Absolute Deviation (3 of 7)
Computational Values for MAD
Forecast Accuracy Mean Absolute Deviation (5 of 7) Can be used to compare accuracy of different forecasting techniques working on the same set of demand data : Exponential smoothing ( = .50): MAD = 4.04
Adjusted exponential smoothing ( = .50, = .30): MAD = 3.81 Linear trend line: MAD = 2.29 Linear trend line has lowest MAD; increasing from .30 to .50 improved smoothed forecast.
Forecast Accuracy Mean Absolute Deviation (6 of 7) A variation on MAD is the mean absolute percent deviation (MAPD). Measures absolute error as a percentage of demand rather than per period.
Eliminates problem of interpreting the measure of accuracy relative to the magnitude of the demand and forecast values. Formula:
Dt Ft 53.41 MAPD .103 or 10.3% 520 Dt
Forecast Accuracy Mean Absolute Deviation (7 of 7) MAPD for other three forecasts: Exponential smoothing ( = .50): MAPD = 8.5% Adjusted exponential smoothing ( = .50, = .30): MAPD = 8.1% Linear trend: MAPD = 4.9%
Forecast Accuracy Cumulative Error (1 of 2) Cumulative error is the sum of the forecast errors (E =ďƒĽet). E = ďƒĽ et = 49.31 indicating forecasts are frequently below actual demand.
Forecast Accuracy Cumulative Error (2 of 2) Cumulative error for other forecasts: Exponential smoothing ( = .50): E = 33.21 Adjusted exponential smoothing ( = .50, =.30): E = 21.14
Average error (bias) is the per period average of cumulative error.
Average error for exponential smoothing forecast: et 49.31 E 4.48 n 11
Forecast Accuracy Example Forecasts by Different Measures
Table 15.9 Comparison of Forecasts for PM Computer Services
Results consistent for all forecasts: Larger value of alpha is preferable. Adjusted forecast is more accurate than exponential smoothing forecasts. Linear trend is more accurate than all the others.
Example
For data below, develop an exponential smoothing forecast using = .40, and an adjusted exponential smoothing forecast using = .40 and = .20. Compare the accuracy of the forecasts using MAD and cumulative error.
Period 1 2 3 4 5 6 7 8
Units 56 61 55 70 66 65 72 75
Step 1: Compute the Exponential Smoothing Forecast. Ft+1 = Dt + (1 - )Ft Step 2: Compute the Adjusted Exponential Smoothing Forecast AFt+1 = Ft +1 + Tt+1 Tt+1 = (Ft +1 - Ft) + (1 - )Tt
Period 1 2 3 4 5 6 7 8 9
Dt 56 61 55 70 66 65 72 75
Ft 56.00 58.00 56.80 62.08 63.65 64.18 67.31 70.39
AFt 56.00 58.40 56.88 63.20 64.86 65.26 68.80 72.19
Dt - Ft Dt - AFt 5.00 5.00 -3.00 -3.40 13.20 13.12 3.92 2.80 1.35 0.14 7.81 6.73 7.68 6.20 35.97 30.60
Step 3: Compute the MAD Values
Dt Ft 41.97 MAD(Ft ) 5.99 n 7 Dt AFt 37.39 MAD( AFt ) 5.34 n 7 Step 4: Compute the Cumulative Error. E(Ft) = 35.97 E(AFt) = 30.60
Problem Statement:
For the following data, Develop a linear regression model Determine a forecast for lumber given 10 building permits in the next quarter.
Quarter 1 2 3 4 5 6 7 8 9 10
Building Permits x 8 12 7 9 15 6 5 8 10 12
Lumber Sales (1,000s of bd ft) y 12.6 16.3 9.3 11.5 18.1 7.6 6.2 14.2 15.0 17.8
Step 1: Compute the Components of the Linear Regression Equation. x 92 92 10 y 128.6 12.86 10 b xy n x y (1,290.3) (10)(9.2)(12.86) 1.25 2 2 ( 932 ) ( 10 )( 9 . 2 ) 2 x n x a y b x 12.86 (1.25)(9.2) 1.36
Step 2: Develop the Linear regression equation. y = a + bx, y = 1.36 + 1.25x
Step 4: Calculate the forecast for x = 10 permits.
Y = a + bx = 1.36 + 1.25(10) = 13.86 or 1,386 board ft