Forecasting Feb 2

3-1

Linear Trend Equation

Ft = Forecast for period t t = Specified number of time periods a = Value of Ft at t = 0 b = Slope of the line

Ft = a + bt

0 1 2 3 4 5 t

Ft

3-2

Calculating a and b

b = n (ty) - t y

n t2 - ( t)2

a = y - b tn

3-3

Linear Trend Equation Example

t yW e e k t 2 S a l e s t y

1 1 1 5 0 1 5 02 4 1 5 7 3 1 43 9 1 6 2 4 8 64 1 6 1 6 6 6 6 45 2 5 1 7 7 8 8 5

t = 1 5 t 2 = 5 5 y = 8 1 2 t y = 2 4 9 9( t ) 2 = 2 2 5

Forecast week 6 & 7 sales?

3-4

Linear Trend Calculation

y = 143.5 + 6.3t

a = 812 - 6.3(15)5 =

b = 5 (2499) - 15(812)5(55) - 225

= 12495-12180275 -225

= 6.3

143.5

Week (t) Sales (y) Sales (y)6 143.5 + 6.3 (6) 181.37 143.5 + 6.3 (7) 187.6

Practice book problem

Calculate Sept forecast using linear trend method

t Y tY From Table 31 with n = 7, t = 28, t2 = 140

50.)28(28)140(7)132(28)542(7

)t(tnYttYnb 22 =

=

=

86.167

)28(50.132n

tbYa ===

1 19 19 2 18 36

3 15 45 4 20 80 5 18 90 6 22 132 7 20 140 28 132 542

Book Problem # 2- Solution

Y= a + bt !For Sept., t = 8, and Yt = 16.86 + .50(8) = 20.86 (000)

1) Linear Trend

7 2011 Pearson Education, Inc. publishing as Prentice Hall

Exponential Smoothing with Trend Adjustment Data

Figure 4.3

| | | | | | | | | 1 2 3 4 5 6 7 8 9

Time (month)

Prod

uct d

eman

d

35 30 25 20 15 10

5 0

Actual demand (At)

There is an upward trend pattern


Exponential Smoothing with Trend Adjustment

When a trend is present, exponential smoothing must be modified

Forecast including (FITt) = trend

Exponentially Exponentially smoothed (Ft) + smoothed (Tt) forecast trend


Exponential Smoothing with Trend Adjustment

Tt = (Ft - Ft - 1) + (1 - )Tt - 1

Step 1: Compute Ft !!Step 2: Compute Tt !!Step 3: Calculate the forecast FITt = Ft + Tt


Exponential Smoothing with Trend Adjustment Example

Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 1 12 11 2 13.00 2 17 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10

Table 4.1



Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 1 12 11 2 13.00 2 17 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10

Table 4.1

F2 = A1 + (1 - )(F1 + T1) F2 = (.2)(12) + (1 - .2)(11 + 2) = 2.4 + 10.4 = 12.8 units

Step 1: Forecast for Month 2



Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 1 12 11 2 13.00 2 17 12.80 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10

Table 4.1

T2 = (F2 - F1) + (1 - )T1 T2 = (.4)(12.8 - 11) + (1 - .4)(2) = .72 + 1.2 = 1.92 units

Step 2: Trend for Month 2



Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 1 12 11 2 13.00 2 17 12.80 1.92 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10

Table 4.1

FIT2 = F2 + T2 FIT2 = 12.8 + 1.92 = 14.72 units

Step 3: Calculate FIT for Month 2



Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 1 12 11 2 13.00 2 17 12.80 1.92 14.72 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10

Table 4.1

! 15.18 2.10 17.28 17.82 2.32 20.14 19.91 2.23 22.14 22.51 2.38 24.89 24.11 2.07 26.18 27.14 2.45 29.59 29.28 2.32 31.60 32.48 2.68 35.16



Figure 4.3

| | | | | | | | | 1 2 3 4 5 6 7 8 9

Time (month)

Prod

uct d

eman

d

35 30 25 20 15 10

5 0

Actual demand (At)

Forecast including trend (FITt) with = .2 and = .4


Seasonal Variations In Data

The multiplicative seasonal model can adjust trend data for seasonal variations in demand


Seasonal Index Example

140 130 120 110 100

90 80 70

| | | | | | | | | | | | J F M A M J J A S O N D

Time

Dem

and

2010 Forecast 2009 Demand 2008 Demand 2007 Demand


Seasonal Variations In Data

1. Find average historical demand for each season 2. Compute the average demand over all seasons 3. Compute a seasonal index for each season 4. Estimate next years total demand 5. Divide this estimate of total demand by the

number of seasons, then multiply it by the seasonal index for that season

Steps in the process:



Jan 80 85 105 90 94 Feb 70 85 85 80 94 Mar 80 93 82 85 94 Apr 90 95 115 100 94 May 113 125 131 123 94 Jun 110 115 120 115 94 Jul 100 102 113 105 94 Aug 88 102 110 100 94 Sept 85 90 95 90 94 Oct 77 78 85 80 94 Nov 75 72 83 80 94 Dec 82 78 80 80 94

Demand Average Average Seasonal Month 2007 2008 2009 2007-2009 Monthly Index

TOTAL 1050 1120 1204 AVE - 93.72



Jan 80 85 105 90 94 Feb 70 85 85 80 94 Mar 80 93 82 85 94 Apr 90 95 115 100 94 May 113 125 131 123 94 Jun 110 115 120 115 94 Jul 100 102 113 105 94 Aug 88 102 110 100 94 Sept 85 90 95 90 94 Oct 77 78 85 80 94 Nov 75 72 83 80 94 Dec 82 78 80 80 94


0.957

Seasonal index = Average 2007-2009 monthly demand Average monthly demand

= 90/94 = .957



Jan 80 85 105 90 94 0.957 Feb 70 85 85 80 94 0.851 Mar 80 93 82 85 94 0.904 Apr 90 95 115 100 94 1.064 May 113 125 131 123 94 1.309 Jun 110 115 120 115 94 1.223 Jul 100 102 113 105 94 1.117 Aug 88 102 110 100 94 1.064 Sept 85 90 95 90 94 0.957 Oct 77 78 85 80 94 0.851 Nov 75 72 83 80 94 0.851 Dec 82 78 80 80 94 0.851




Jan 80 85 105 90 94 0.957 Feb 70 85 85 80 94 0.851 Mar 80 93 82 85 94 0.904 Apr 90 95 115 100 94 1.064 May 113 125 131 123 94 1.309 Jun 110 115 120 115 94 1.223 Jul 100 102 113 105 94 1.117 Aug 88 102 110 100 94 1.064 Sept 85 90 95 90 94 0.957 Oct 77 78 85 80 94 0.851 Nov 75 72 83 80 94 0.851 Dec 82 78 80 80 94 0.851


Expected annual demand = 1,200

Jan x .957 = 96 1,200 12

Feb x .851 = 85 1,200 12

Forecast for 2010



140 130 120 110 100

90 80 70

| | | | | | | | | | | | J F M A M J J A S O N D

Time

Dem

and

2010 Forecast 2009 Demand 2008 Demand 2007 Demand

24 2011 Pearson Education, Inc.

Associative ForecastingUsed when changes in one or more independent variables can be used to predict the changes in the dependent variable. Some examples !-Sales of mountain bikes may be related to the percentage of the young population living in that area. -Ice cream sales can be related to temperature - Increase in fuel cost leads to price increases in

products and services !

!!!Most common technique is linear regression analysis same technique just as we did in the time series example


Associative ForecastingForecasting an outcome based on predictor variables using the least squares technique

y = a + bx^

where y = computed value of the variable to be predicted (dependent variable)

a = y-axis intercept b = slope of the regression line x = the independent variable though to

predict the value of the dependent variable

^


Associative Forecasting Example

Sales Area Payroll ($ millions), y ($ billions), x 2.0 1 3.0 3 2.5 4 2.0 2 2.0 1 3.5 7

4.0 3.0 2.0 1.0

| | | | | | | 0 1 2 3 4 5 6 7

Sale

s

Area payroll

Forecast sales amount for $ 6B???



Sales, y Payroll, x x2 xy 2.0 1 1 2.0 3.0 3 9 9.0 2.5 4 16 10.0 2.0 2 4 4.0 2.0 1 1 2.0 3.5 7 49 24.5 y = 15.0 x = 18 x2 = 80 xy = 51.5

x = x/6 = 18/6 = 3

y = y/6 = 15/6 = 2.5

b = = = .25xy - nxy x2 - nx2

51.5 - (6)(3)(2.5) 80 - (6)(32)

a = y - bx = 2.5 - (.25)(3) = 1.75



y = 1.75 + .25x^ Sales = 1.75 + .25(payroll)

If payroll next year is estimated to be $6 billion, then:

Sales = 1.75 + .25(6) Sales = $3,250,000

4.0 3.0 2.0 1.0

| | | | | | | 0 1 2 3 4 5 6 7

Nod

els

sal

es

Area payroll

3.25

3-29

Forecast Accuracy Error: difference between actual value and

predicted value Mean Absolute Deviation (MAD)

Average absolute error

Mean Squared Error (MSE) Average of squared error

Mean Absolute Percent Error (MAPE) Average absolute percent error

3-30

MAD, MSE, and MAPE

MAD =Actual forecast

n

MSE = Actual forecast)

-1

2

n

(

MAPE = Actualforecast

n

/ Actual*100)(

3-31

MAD, MSE, and MAPE

MAD Easy to compute Weights errors linearly

MSE Squares error More weight to large errors

MAPE Puts errors in perspective

3-32

Example 10

Period Actual Forecast (A-F) |A-F| (A-F)^2 (|A-F|/Actual)*1001 217 215 2 2 4 0.922 213 216 -3 3 9 1.413 216 215 1 1 1 0.464 210 214 -4 4 16 1.905 213 211 2 2 4 0.946 219 214 5 5 25 2.287 216 217 -1 1 1 0.468 212 216 -4 4 16 1.89

-2 22 76 10.26

MAD= 2.75 (22 / 8 )MSE= 10.86 ( 76 / 7 )MAPE= 1.28 (10.26 / 8)

3-33

Tracking Signal

Tracking signal = (Actual-forecast)MAD

Tracking signal Ratio of cumulative error to MAD

Bias: Persistent tendency for forecasts to be greater or less than actual values.

3-34

Choosing a Forecasting Technique

No single technique works in every situation Two most important factors

Cost Accuracy

Other factors include the availability of: Historical data Computers Time needed to gather and analyze the data Forecast horizon

Documents

Forecasting Feb 2