Exponential Smoothing with Trend. As we move toward medium-range forecasts, trend becomes more important.

(1)

Exponential Smoothing with Trend

• As we move toward medium-range

forecasts, trend becomes more important.

• Incorporating a trend component into

exponentially smoothed forecasts is called double exponential smoothing.

– The estimate for the average and the estimate

for the trend are both smoothed.

(2)

Exponential Smoothing with Trend Adjustment

Forecast including trend (FIT

_t

)OR Adjusted Forecast (AF

_t

) = exponentially smoothed forecast (F

_t

) + exponentially smoothed trend (T

_t

)

That is, AF

_t

= F

_t

+ T

_t

We need to compute both Ft and Tt

(3)

F

_t

= Last period’s forecast

+ D (Last period’s actual –Last period’s forecast) F

_t

= F

_t-1

+ D (A

_t-1

–F

_t-1

)

or

T

_t

= E(This period’s Forecast - last period’s Forecast) + (1- E) (Trend estimate last period)

T

_t

= E(F

_t

- F

_t-1

) + (1- E) T

_t-1

for all t

or

Exponential Smoothing with Trend Adjustment – (contd.)

F

_t

= exponentially smoothed forecast of the data series in period t T

_t

= exponentially smoothed trend in period t

A

_t

= actual demand in period t

α = smoothing constant for the average

β = smoothing constant for the trend

(4)

Adjusted Exponential Smoothing Example

PERIOD MONTH DEMAND

1 Jan 37

2 Feb 40

3 Mar 41

4 Apr 37

5 May 45

6 Jun 50

7 Jul 43

8 Aug 47

9 Sep 56

10 Oct 52

11 Nov 55

12 Dec 54

(5)

Adjusted Exponential Smoothing Example

Per Month Dem F_t+1 T_t+1 AF_t+1

1 Jan 37 37.00 - -

2 Feb 40 37.00 0.00 37.00

3 Mar 41 38.50 0.45 38.95

4 Apr 37 39.75 0.69 40.44

5 May 45 38.37 0.07 38.44

6 Jun 50 41.68 1.04 42.73

7 Jul 43 45.84 1.97 47.82

8 Aug 47 44.42 0.95 45.37

9 Sep 56 45.71 1.05 46.76

10 Oct 52 50.85 2.28 58.13

11 Nov 55 51.42 1.76 53.19

12 Dec 54 53.21 1.77 54.98

13 Jan - 53.61 1.36 54.96

F₃=F₂+0.50(A₂-F₂) = 37+0.50*3 = 38.5 T₃ = E(F₃- F₂) + (1 - E) T₂

= (0.30)(38.5 - 37.0) + (0.70)(0)

= 0.45

AF₃ = F₃ + T₃

=38.5 + 0.45 = 38.95 T₁₃ = E(F₁₃- F₁₂) + (1 - E) T₁₂

=(0.30)(53.61 - 53.21) + (0.70)(1.77)

=1.36

AF₁₃ = F₁₃ + T₁₃= 53.61 + 1.36 = 54.96

Forecast ( D = 0.50)

(6)

Adjusted Exponential Smoothing Forecasts

70 – 60 – 50 – 40 – 30 – 20 – 10 –

0 – | | | | | | | | | | | | |

1 2 3 4 5 6 7 8 9 10 11 12 13

Adjusted forecast (D = 0.50; E = 0.30)) Actual

Demand

Period

Forecast (D = 0.50)

(7)

Seasonal Adjustments

i i

S D D

Seasonal factor = ⁼ ∑

• Repetitive increase/decrease in demand

– Use seasonal factor to adjust forecast

Where

D

_i

is the sum of demands of the period i in the time series data

6Dis net sum of demands of the entire period in the time series data

(8)

Example: Seasonal Adjustment [1]

Demand (1000’s per quarter)

Year 1 2 3 4 Total

2004 12.6 8.6 6.3 17.5 45.0

2005 14.1 10.3 7.5 18.2 50.1

2006 15.3 10.6 8.1 19.6 53.6

Total 42.0 29.5 21.9 55.3 148.7

S_i 0.28 0.20 0.15 0.37

Computed trend line for data y = 40.97 + 4.30 X [Given to you]

2006 (year 4) forecast = 40.97 + 4.30 (4) = 58.17

Forecasted demand after seasonal adjustment for the year 2006 is

2006 16.28 11.63 8.73 21.53

--- Details

58.17 x 0.28 = 16.28; 58.17 x 0.20 = 11.63;

58.17 x 0.15 = 8.73; 58.17 x 0.37 = 21.53

---

1 1 42 0

148 7 0 28

S D

= D = =

∑

^.^. ^.

SF

₁

= (S

₁

) (F

₅

)

= (0.28)(58.17) = 16.28 SF

₂

= (S

₂

) (F

₅

)

= (0.20)(58.17) = 11.63 SF

₃

= (S

₃

) (F

₅

)

= (0.15)(58.17) = 8.73 SF

₄

= (S

₄

) (F

₅

)

= (0.37)(58.17) = 21.53

(9)

Quarter Year 1 Year 2 Year 3 Year 4

1 45 70 100 100

2 335 370 585 725

3 520 590 830 1160

4 100 170 285 215

Total 1000 1200 1800 2200

Average 250 300 450 550

Example: Seasonal Adjustment [2]

(10)

1 45 70 100 100

2 335 370 585 725

3 520 590 830 1160

4 100 170 285 215

Total 1000 1200 1800 2200

Average 250 300 450 550

Seasonal Index = Actual Demand Average Demand

Example: Seasonal Adjustment [2]

(11)

1 45 70 100 100

2 335 370 585 725

3 520 590 830 1160

4 100 170 285 215

Total 1000 1200 1800 2200

Average 250 300 450 550

Seasonal Index = = 0.18 45 250

Example: Seasonal Adjustment [2]

(12)

1 45/250 = 0.18 70 100 100

2 335 370 585 725

3 520 590 830 1160

4 100 170 285 215

Total 1000 1200 1800 2200

Average 250 300 450 550

Seasonal Index = = 0.18 45 250

Example: Seasonal Adjustment [2] – Contd.

(13)

Quarter Year 1 Year 2 Year 3 Year 4 1 45/250 = 0.18 70/300 = 0.23 100/450 = 0.22 100/550 = 0.18 2 335/250 = 1.34 370/300 = 1.23 585/450 = 1.30 725/550 = 1.32 3 520/250 = 2.08 590/300 = 1.97 830/450 = 1.84 1160/550 = 2.11 4 100/250 = 0.40 170/300 = 0.57 285/450 = 0.63 215/550 = 0.39

Example: Seasonal Adjustment [2] – Contd.

(14)

Quarter Average Seasonal Index

1 (0.18 + 0.23 + 0.22 + 0.18)/4 = 0.20 2 3

4 Example: Seasonal Adjustment [2] – Contd.

(15)

Quarter Average Seasonal Index

1 (0.18 + 0.23 + 0.22 + 0.18)/4 = 0.20 2 (1.34 + 1.23 + 1.30 + 1.32)/4 = 1.30 3 (2.08 + 1.97 + 1.84 + 2.11)/4 = 2.00 4 (0.40 + 0.57 + 0.63 + 0.39)/4 = 0.50

Example: Seasonal Adjustment [2] – Contd.

(16)

Quarter Average Seasonal Index Forecast 1 (0.18 + 0.23 + 0.22 + 0.18)/4 = 0.20

2 (1.34 + 1.23 + 1.30 + 1.32)/4 = 1.30 3 (2.08 + 1.97 + 1.84 + 2.11)/4 = 2.00 4 (0.40 + 0.57 + 0.63 + 0.39)/4 = 0.50

Projected Annual Demand = 2600 [Given]

Average Quarterly Demand = 2600/4 = 650

Example: Seasonal Adjustment [2] – Contd.

(17)

Quarter Average Seasonal Index Forecast

1 (0.18 + 0.23 + 0.22 + 0.18)/4 = 0.20 650(0.20) = 130 2 (1.34 + 1.23 + 1.30 + 1.32)/4 = 1.30

3 (2.08 + 1.97 + 1.84 + 2.11)/4 = 2.00 4 (0.40 + 0.57 + 0.63 + 0.39)/4 = 0.50

Projected Annual Demand = 2600

Average Quarterly Demand = 2600/4 = 650

Example: Seasonal Adjustment [2] – Contd.

(18)

Quarter Average Seasonal Index Forecast

1 (0.18 + 0.23 + 0.22 + 0.18)/4 = 0.20 650(0.20) = 130 2 (1.34 + 1.23 + 1.30 + 1.32)/4 = 1.30 650(1.30) = 845 3 (2.08 + 1.97 + 1.84 + 2.11)/4 = 2.00 650(2.00) = 1300 4 (0.40 + 0.57 + 0.63 + 0.39)/4 = 0.50 650(0.50) = 325

Example: Seasonal Adjustment [2] – Contd.

(19)

Seasonalised Time Series Regression Analysis

1. Select a representative historical data set.

2. Develop a seasonal index for each season.

3. Use the seasonal indexes to De-Seasonalise the data.

4. Perform linear regression analysis on the De-Seasonalised data.

5. Use the regression equation to compute the forecasts.

6. Use the seasonal indexes to reapply the seasonal patterns to the

forecasts.

(20)

Example: Computer Products Corp.

Seasonalized Times Series Regression Analysis

An analyst at CPC wants to develop next year’s

quarterly forecasts of sales revenue for CPC’s line

of Epsilon Computers. She believes that the most

recent 8 quarters of sales (shown on the next slide)

are representative of next year’s sales.

(21)

Example: Computer Products Corp.

• Seasonalised Times Series Regression Analysis – Representative Historical Data Set

Year Qtr. ($mil.) Year Qtr. ($mil.)

1 1 7.4 2 1 8.3

1 2 6.5 2 2 7.4

1 3 4.9 2 3 5.4

1 4 16.1 2 4 18.0

(22)

Example: Computer Products Corp.

– Compute the Seasonal Indexes

Quarterly Sales

Year Q1 Q2 Q3 Q4 Total

1 7.4 6.5 4.9 16.1 34.9

2 8.3 7.4 5.4 18.0 39.1

Totals 15.7 13.9 10.3 34.1 74.0 Qtr. Avg. 7.85 6.95 5.15 17.05 9.25 Seas.Ind. .849 .751 .557 1.843 4.000

7.85 / 9.25

(23)

Example: Computer Products Corp.

De-Seasonalise the Data

Quarterly Sales

Year Q1 Q2 Q3 Q4

1 8.72 8.66 8.80 8.74

2 9.78 9.85 9.69 9.77

De-Seasonalised data for Q1 = { Actual Q1 sales / Seas. Index } Time series data:

Quarterly Sales

Year Q1 Q2 Q3 Q4

1 7.4 6.5 4.9 16.1

2 8.3 7.4 5.4 18.0

Seasonal Index 0.849 0.751 0.557 1.843

Y

_t

= T

_t

x S

_t

x C

_t

x R

_t

Assum. There is no C_t& R_t

Y

_t

= T

_t

x S

_t

Y

_t

(dese.)= (Y

_t

/S

_t

)

(24)

Example: Computer Products Corp.

– Perform Regression on De-seasonalized Data

Yr. Qtr. x y x

²

xy

1 1 1 8.72 1 8.72

1 2 2 8.66 4 17.32

1 3 3 8.80 9 26.40

1 4 4 8.74 16 34.96

2 1 5 9.78 25 48.90

2 2 6 9.85 36 59.10

2 3 7 9.69 49 67.83

2 4 8 9.77 64 78.16

Totals 36 74.01 204 341.39

Y = 8.357 + 0.199X

2

204(74.01) 36(341.39 )

a 8.357

8(204) (36 )

= − =

−

²

8(341.39) 36(74.01)

b 0.199

8(204) (36)

= − =

−

(25)

Example: Computer Products Corp.

Y

₉

= 8.357 + 0.199(9) = 10.148 Y

₁₀

= 8.357 + 0.199(10) = 10.347 Y

₁₁

= 8.357 + 0.199(11) = 10.546 Y

₁₂

= 8.357 + 0.199(12) = 10.745

Note: Average sales are expected to increase by .199 million (about $200,000) per quarter.

MODEL : Y = 8.357 + 0.199X

Compute the De-Seasonalised Forecasts

(26)

Example: Computer Products Corp.

Seasonalised the Forecasts

Seas. De-seas. Seas.

Yr. Qtr. Index Forecast Forecast

3 1 .849 10.148 8.62

3 2 .751 10.347 7.77

3 3 .557 10.546 5.87

3 4 1.843 10.745 19.80

(27)

Time Series Models & Classical Decomposition

• Decomposition time series models:

• Multiplicative: Y = T x C x S x e

• Additive: Y = T + C + S + e

T = Trend component C = Cyclical component S = Seasonal component

e = Error or random component

(28)

Time Series Models & Classical Decomposition

• Classical decomposition is used to isolate trend,

seasonal, and other variability components from a

time series model

(29)

Classical Decomposition Explained

Basic Steps:

1. Determine seasonal indexes using the ratio to moving average method

2. Deseasonalize the data

3. Develop the trend-cyclical regression equation using deseasonalized data

4. Multiply the forecasted trend values by their

seasonal indexes to create a more accurate

forecast

(30)

• Start with multiplicative model…

Y = TCSe

• Then

Se = (Y/TC)

(31)

Classical Decomposition:

Illustration

• Gem Company’ s operations department has been asked to deseasonalize and forecast sales for the next four quarters of the coming year

• The Company has compiled its past sales data in Table 1

• An illustration using classical decomposition will follow

Table 1: Gem Company’s Sales Data

Original Forecasted Year Quarter Period Sales Sales

t Y TS

1 1 1 55 -

2 2 47 -

3 3 65 -

4 4 70 -

2 1 5 65 -

2 6 58 -

3 7 75 -

4 8 80 -

3 1 9 65 -

2 10 62 -

3 11 80 -

4 12 85 -

4 1 13 70 -

2 14 65 -

3 15 85 -

4 16 90 -

5 1 17 - ?

2 18 - ?

3 19 - ?

4 20 - ?

(32)

Classical Decomposition Illustration:

Step 1

• (a) Compute the four- quarter simple

moving average

Ex: simple MA at end of Qtr 2 and

beginning of Qtr 3 (55+47+65+70)/4 =

59.25

Moving Year Quarter Period Sales Average

t Y

1 1 1 55

2 2 47 59.25

3 3 65 61.75

4 4 70 64.50

2 1 5 65 67.00

2 6 58 69.50

3 7 75 69.50

4 8 80 70.50

3 1 9 65 71.75

2 10 62 73.00

3 11 80 74.25

4 12 85 75.00

4 1 13 70 76.25

2 14 65 77.50

3 15 85

4 16 90

(33)

Classical Decomposition Illustration: ^{Step 1}

• (b) Compute the two- quarter centered

moving average

Ex: centered MA at middle of Qtr 3

(59.25+61.25)/2

= 60.500

Table 2: Four-Quarter Moving Average

Simple Centered Moving Moving Year Quarter Period Sales Average Average

t Y TC

1 1 1 55

2 2 47 59.25

3 3 65 61.75 60.500

4 4 70 64.50 63.125

2 1 5 65 67.00 65.750

2 6 58 69.50 68.250

3 7 75 69.50 69.500

4 8 80 70.50 70.000

3 1 9 65 71.75 71.125

2 10 62 73.00 72.375

3 11 80 74.25 73.625

4 12 85 75.00 74.625

4 1 13 70 76.25 75.625

2 14 65 77.50 76.875

3 15 85

4 16 90

(34)

Classical Decomposition Illustration: ^{Step 1}

• (c) Compute the seasonal-error

component (percent MA)

Ex: percent MA at Qtr 3

(65/60.500)

= 1.074

Table 2: Four-Quarter Moving Average

Simple Centered Percent Moving Moving Moving Year Quarter Period Sales Average Average Average

t Y TC Se=Y/(TC)

1 1 1 55

2 2 47 59.25

3 3 65 61.75 60.500 1.074

4 4 70 64.50 63.125 1.109

2 1 5 65 67.00 65.750 0.989

2 6 58 69.50 68.250 0.850

3 7 75 69.50 69.500 1.079

4 8 80 70.50 70.000 1.143

3 1 9 65 71.75 71.125 0.914

2 10 62 73.00 72.375 0.857

3 11 80 74.25 73.625 1.087

4 12 85 75.00 74.625 1.139

4 1 13 70 76.25 75.625 0.926

2 14 65 77.50 76.875 0.846

3 15 85

4 16 90

(35)

Classical Decomposition Illustration: ^{Step 1}

• (d) Compute the unadjusted seasonal index using the seasonal- error components from Table 2

Ex (Qtr 1): [(Yr 2, Qtr 1) + (Yr 3, Qtr 1) + (Yr 4, Qtr 1)]/3

= [0.989+0.914+0.926]/3 = 0.943

Table 3: Seasonal Index Computation

Unadjusted Adjusted

Seasonal Adjusting Seasonal

Quarter Average Index Factor Index

1 (0.989+0.914+0.926)/3 = 0.943 x (4.000/4.004) = 0.942 2 (0.850+0.857+0.846)/3 = 0.851 x (4.000/4.004) = 0.850 3 (1.074+1.079+1.087)/3 = 1.080 x (4.000/4.004) = 1.079 4 (1.109+1.143+1.139)/3 = 1.130 x (4.000/4.004) = 1.129

4.004 4.000

(36)

Classical Decomposition Illustration: ^{Step 1}

• (e) Compute the adjusting factor by dividing the number of quarters (4) by the sum of all calculated unadjusted seasonal indexes

= 4.000/(0.943+0.851+1.080+1.130) = (4.000/4.004)

1 (0.989+0.914+0.926)/3 = 0.943 x (4.000/4.004) = 0.942 2 (0.850+0.857+0.846)/3 = 0.851 x (4.000/4.004) = 0.850 3 (1.074+1.079+1.087)/3 = 1.080 x (4.000/4.004) = 1.079 4 (1.109+1.143+1.139)/3 = 1.130 x (4.000/4.004) = 1.129

4.004 4.000

(37)

Classical Decomposition Illustration: ^{Step 1}

• (f) Compute the adjusted seasonal index by multiplying the unadjusted seasonal index by the adjusting factor

Ex (Qtr 1): 0.943 x (4.000/4.004) = 0.942

1 (0.989+0.914+0.926)/3 = 0.943 x (4.000/4.004) = 0.942 2 (0.850+0.857+0.846)/3 = 0.851 x (4.000/4.004) = 0.850 3 (1.074+1.079+1.087)/3 = 1.080 x (4.000/4.004) = 1.079 4 (1.109+1.143+1.139)/3 = 1.130 x (4.000/4.004) = 1.129

4.004 4.000

(38)

Classical Decomposition Illustration: ^{Step 2}

• Compute the

deseasonalized sales by dividing original sales by the adjusted seasonal index

Ex (Yr 1, Qtr 1):

(55 / 0.942)

= 58.386

Table 4: Deseasonalizing Sales

Adjusted

Original Seasonal Deseasonalized Year Quarter Period Sales Index Sales

t Y S TCe

1 1 1 55 0.942 58.386

2 2 47 0.850 55.294

3 3 65 1.079 60.241

4 4 70 1.129 62.002

2 1 5 65 0.942 69.002

2 6 58 0.850 68.235

3 7 75 1.079 69.509

4 8 80 1.129 70.859

3 1 9 65 0.942 69.002

2 10 62 0.850 72.941

3 11 80 1.079 74.143

4 12 85 1.129 75.288

4 1 13 70 0.942 74.310

2 14 65 0.850 76.471

3 15 85 1.079 78.777

4 16 90 1.129 79.717

(39)

Classical Decomposition Illustration: ^{Step 3}

• Compute the trend-cyclical regression equation using simple linear regression T

_t

= a + bt

t-bar = 8.5 T-bar = 69.6

b = 1.465

a = 57.180 T

_t

= 57.180 + 1.465t

Table 5: Regression Equation Values

Deseasonalized Year Quarter Period Sales

t TCe = (Y/S) t(Y/S) t²

1 1 1 58.386 58.386 1

2 2 55.294 110.588 4

3 3 60.241 180.723 9

4 4 62.002 248.007 16

2 1 5 69.002 345.011 25

2 6 68.235 409.412 36

3 7 69.509 486.562 49

4 8 70.859 566.873 64

3 1 9 69.002 621.019 81

2 10 72.941 729.412 100

3 11 74.143 815.570 121

4 12 75.288 903.454 144

4 1 13 74.310 966.030 169

2 14 76.471 1070.588 196

3 15 78.777 1181.650 225

4 16 79.717 1275.465 256

136 1114.176 9968.750 1496

(40)

Classical Decomposition Illustration: ^{Step 4}

• (a) Develop trend sales T

_t

= 57.180 + 1.465t Ex (Yr 1, Qtr 1):

T

₁

= 57.180 + 1.465(1)

= 58.645

Table 6: Trend Sales

Original Deseasonalized Trend Year Quarter Period Sales Sales Sales

t Y TCe = (Y/S) T

1 1 1 55 58.386 58.645

2 2 47 55.294 60.110

3 3 65 60.241 61.575

4 4 70 62.002 63.040

2 1 5 65 69.002 64.505

2 6 58 68.235 65.970

3 7 75 69.509 67.435

4 8 80 70.859 68.900

3 1 9 65 69.002 70.365

2 10 62 72.941 71.830

3 11 80 74.143 73.295

4 12 85 75.288 74.760

4 1 13 70 74.310 76.225

2 14 65 76.471 77.690

3 15 85 78.777 79.155

4 16 90 79.717 80.620

5 1 17 82.085

2 18 83.550

3 19 85.015

4 20 86.480

(41)

Classical Decomposition Illustration: ^{Step 4}

• (b) Forecast sales for each of the four quarters of the coming year

Ex (Yr 5, Qtr 1):

0.942 x 82.085

= 77.324

Table 7: Forecasted Sales

Seasonal Trend Forecasted Year Quarter Period Index Sales Sales

t S T TS

1 1 1 0.942 58.645

2 2 0.850 60.110

3 3 1.079 61.575

4 4 1.129 63.040

2 1 5 0.942 64.505

2 6 0.850 65.970

3 7 1.079 67.435

4 8 1.129 68.900

3 1 9 0.942 70.365

2 10 0.850 71.830

3 11 1.079 73.295

4 12 1.129 74.760

4 1 13 0.942 76.225

2 14 0.850 77.690

3 15 1.079 79.155

4 16 1.129 80.620

5 1 17 0.942 82.085 77.324

2 18 0.850 83.550 71.018

3 19 1.079 85.015 91.731

4 20 1.129 86.480 97.636

(42)

Classical Decomposition Illustration: Graphical Look

Graph 1: Comparison of Trend, Original, and Deseasonalized Sales

40 50 60 70 80 90 100

0 2 4 6 8 10 12 14 16 18

Quarter Sales ($) (Y/S) = TCe

Deseasonalized

Y Original T

Trend

(43)

• Smooth the time series to remove

random effects and seasonality. • Calculate moving averages.

• Determine “period factors” to isolate the (seasonal)

•

(error)

factor. • Calculate the ratio y

_t

/MA

_t

.

• Determine the “unadjusted

seasonal factors” to eliminate the random component from the period factors

The Classical Decomposition- Procedure

• Average all the y

_t

/MA

_t

that

correspond to the same season.

(44)

• Determine the “adjusted seasonal factors”.

Calculate:

[Unadjusted seasonal factor]

[Average seasonal factor]

• Determine “Deseasonalized data

values”. Calculate:

y

_t

[Adjusted seasonal factors]

_t

• Determine a deseasonalized trend forecast.

The Classical Decomposition- Procedure – Contd.

Use linear regression on the deseasonalized time series.

Calculate:

(y

_t

/Ma

_t

)

•

[Adjusted seasonal forecast].

• Determine an “adjusted seasonal

forecast”.

(45)

Monitoring and Controlling Operations Forecasts

• Reasons for out-of-control forecasts

– change in trend

– appearance of cycle – politics

– weather changes

– promotions

(46)

Monitoring and Controlling a Forecasting Model

Forecasts can be monitored using either Tracking Signal (TS) or Control Charts

• Why track the forecast?

– To plan around the error in the future

– To measure actual demand versus forecasts

– To improve our forecasting methods

(47)

Monitoring and Controlling a Forecasting Model

• Tracking Signal (TS)

– The TS measures the cumulative forecast error over n periods in terms of MAD

– If the forecasting model is performing well, the TS should be around zero

– TS indicates the direction of the forecasting error

• if the TS is positive -- increase the forecasts,

• if the TS is negative -- decrease the forecasts.

n

i i

1

(Actual demand - Forecast demand ) TS =

MAD

∑

i=

(48)

Monitoring and Controlling a Forecasting Model

• Tracking Signal

– The value of the TS can be used to automatically trigger new parameter values of a model, thereby correcting model performance.

– If the limits are set too narrow, the parameter values will be changed too often.

– If the limits are set too wide, the parameter values will

not be changed often enough and accuracy will suffer.

(49)

Mo Fcst Act Error RSFE Abs

Error Cum MAD TS

1 100 90 2 100 95 3 100 115 4 100 100 5 100 125 6 100 140

|Error|

Tracking Signal Computation

(50)

Mo Forc Act Error RSFE Abs

Error Cum MAD TS

1 100 90 2 100 95 3 100 115 4 100 100 5 100 125 6 100 140

-10

Error = Actual - Forecast

= 90 - 100 = -10

|Error|

Tracking Signal Computation

(51)

Mo Forc Act Error RSFE Abs

Error Cum MAD TS

1 100 90 2 100 95 3 100 115 4 100 100 5 100 125 6 100 140

-10 -10

RSFE = 6 Errors

= NA + (-10) = -10

|Error|

Tracking Signal Computation

(52)

Mo Forc Act Error RSFE Abs

Error Cum MAD TS

1 100 90 2 100 95 3 100 115 4 100 100 5 100 125 6 100 140

-10 -10 10

Abs Error = |Error|

= |-10| = 10

|Error|

Tracking Signal Computation

(53)

Mo Forc Act Error RSFE Abs

Error Cum MAD TS

1 100 90 2 100 95 3 100 115 4 100 100 5 100 125 6 100 140

-10 -10 10 10

Cum |Error| = 6 |Errors|

= NA + 10 = 10

|Error|

Tracking Signal Computation

(54)

Mo Forc Act Error RSFE Abs

Error Cum

|Error| MAD TS

1 100 90 2 100 95 3 100 115 4 100 100 5 100 125 6 100 140

-10 -10 10 10 10.0

MAD = 6 |Errors|/n

= 10/1 = 10

Tracking Signal Computation

(55)

Mo Forc Act Error RSFE Abs

Error Cum MAD TS

1 100 90 2 100 95 3 100 115 4 100 100 5 100 125 6 100 140

-10 -10 10 10 10.0 -1

TS = RSFE/MAD

= -10/10 = -1

|Error|

Tracking Signal Computation

(56)

Mo Forc Act Error RSFE Abs

Error Cum MAD TS

1 100 90 2 100 95 3 100 115 4 100 100 5 100 125 6 100 140

-10 -10 10 10 10.0 -1 -5

Error = Actual - Forecast

= 95 - 100 = -5

|Error|

Tracking Signal Computation

(57)

Mo Forc Act Error RSFE Abs

Error Cum MAD TS

1 100 90 2 100 95 3 100 115 4 100 100 5 100 125 6 100 140

-10 -10 10 10 10.0 -1 -5 -15

RSFE = 6 Errors

= (-10) + (-5) = -15

|Error|

Tracking Signal Computation

(58)

Mo Forc Act Error RSFE Abs

Error Cum MAD TS

1 100 90 2 100 95 3 100 115 4 100 100 5 100 125 6 100 140

-10 -10 10 10 10.0 -1 -5 -15 5

Abs Error = |Error|

= |-5| = 5

|Error|

Tracking Signal Computation

(59)

Mo Forc Act Error RSFE Abs

Error Cum MAD TS

1 100 90 2 100 95 3 100 115 4 100 100 5 100 125 6 100 140

-10 -10 10 10 10.0 -1

-5 -15 5 15

Cum Error = 6 |Errors|

= 10 + 5 = 15

|Error|

Tracking Signal Computation

(60)

Mo Forc Act Error RSFE Abs

Error Cum MAD TS

1 100 90 2 100 95 3 100 115 4 100 100 5 100 125 6 100 140

-10 -10 10 10 10.0 -1 -5 -15 5 15 7.5

MAD = 6 |Errors|/n

= 15/2 = 7.5

|Error|

Tracking Signal Computation

(61)

Mo Forc Act Error RSFE Abs

Error Cum MAD TS

1 100 90 2 100 95 3 100 115 4 100 100 5 100 125 6 100 140

-10 -10 10 10 10.0 -1 -5 -15 5 15 7.5 -2

|Error|

TS = RSFE/MAD

= -15/7.5 = -2

Tracking Signal Computation

(62)

Plot of a Tracking Signal

Time Lower control limit Upper control limit

Signal exceeded limit

Tracking signal

Acceptable range

M AD

+

0 -

(63)

Tracking Signals

0 20 40 60 80 100 120 140 160

0 1 2 3 4 5 6 7

Time

Actual Demand

-3 -2 -1 0 1 2 3

Tracking Singal

Tracking Signal Forecast

Actual demand

(64)

¾ The cumulative forecast error reflects the bias in forecasts, which is the persistent tendency for forecasts to be greater or less than the actual values of a time series.

¾ Tracking signal values are compared to predetermined limits

based on judgment and experience. They often range from r3 to r8; for the most part, we shall use limits of ±4, which are roughly comparable to three standard deviation limits.

¾ Values within the limits suggest –but do not guarantee–that the forecast is performing adequately.

NOTE on TS

(65)

Statistical Control Charts

V = ¦(D

_t

- F

_t

)

²

n - 1

9 This methods assumes (a) Forecast errors are randomly distributed around a mean of zero and (b) The

distribution of errors is normal.

The control chart approach involves setting upper and lower limits for individual forecast errors (instead of cumulative errors, as in the case with a tracking signal). The limits are multiples of the “square root of MSE”

(The square root of MSE is used in practice as an estimate of the standard deviation, V , of the distribution of errors).

9 Using V we can calculate statistical control limits for the

forecast error

(66)

Statistical Control Charts (Contd.)

9 Recall that for a ND, approximately 95% of the values (errors in this case) can be expected to fall within limits of 0 r 2V, and approximately 99.7% of the values can be expected to fall within r 3V of zero.

9 Hence, if the forecast is “in control”, 99.7% or 95% of the errors should fall within the limits, depending upon whether r 3V or r 2V limits are used.

9 Points that fall outside these limits should be regarded as

evidence that corrective action is needed [that is the forecast is

not performing adequately).

(67)

Statistical Control Charts

Errors

18.39 – 12.24 – 6.12 – 0 – -6.12 – -12.24 – -18.39 –

| | | | | | | | | | | | |

0 1 2 3 4 5 6 7 8 9 10 11 12

Period

(68)

Statistical Control Charts

Errors

18.39 – 12.24 – 6.12 – 0 – -6.12 – -12.24 – -18.39 –

| | | | | | | | | | | | |

0 1 2 3 4 5 6 7 8 9 10 11 12

Period UCL = +3V

LCL = -3V

(69)

Ranging Forecasts

• Forecasts for future periods are only estimates and are subject to error.

– One way to deal with uncertainty is to develop best-estimate forecasts and the ranges within which the actual data are likely to fall.

• The ranges of a forecast are defined by the upper

and lower limits of a confidence interval.

(70)

Ranging Forecasts

• The ranges or limits of a forecast are estimated by:

Upper limit = Y + t(s

_yx

) Lower limit = Y - t(s

_yx

) where:

Y = best-estimate forecast

t = number of standard deviations from the mean

of the distribution to provide a given probability of exceeding the limits through chance

s

_yx

= standard error of the forecast

(71)

Ranging Forecasts

• The standard error (deviation) of the forecast is computed as:

2 yx

y - a y - b xy s =

n - 2

∑ ∑ ∑

(72)

Example: Railroad Products Co.

• Ranging Forecasts

Recall that linear regression analysis

provided a forecast of annual sales for RPC in year 8 equal to $20.55 million.

Set the limits (ranges) of the forecast so

that there is only a 5 percent probability of

exceeding the limits by chance.

(73)

Example: Railroad Products Co.

• Ranging Forecasts

Step 1: Compute the standard error of the forecasts, s

_yx

.

Step 2: Determine the appropriate value for t.

n = 7, so degrees of freedom = n – 2 = 5.

Area in upper tail = .05/2 = .025 Statistical Table shows t = 2.571.

1287.5 .528(93) .0801(15, 440)

.5748

yx

7 2

s = − − =

−

(74)

Example: Railroad Products Co.

• Ranging Forecasts

– Step 3: Compute upper and lower limits.

Upper limit = 20.55 + 2.571(.5748)

= 20.55 + 1.478

= 22.028

Lower limit = 20.55 - 2.571(.5748)

= 20.55 - 1.478

= 19.072

We are 95% confident that the actual sales for year

8 will be between $19.072 and $22.028 million.

(75)

Criteria/factor to be considered for Selecting a Forecasting Method

• Cost

• Accuracy

• Data available

• Time span

• Nature of products and services

• Impulse response and noise dampening

(76)

• Cost and Accuracy

– There is a trade-off between cost and accuracy;

generally, more forecast accuracy can be obtained at a cost.

– High-accuracy approaches have disadvantages :

• Use more data

• Data are ordinarily more difficult to obtain

• The models are more costly to design, implement, and operate

• Take longer to use

Criteria for Selecting a Forecasting Method

- Low/Moderate-Cost Approaches – statistical models, historical analogies, executive-committee consensus - High-Cost Approaches – complex econometric models,

Delphi, and market research

(77)

• Availability of historical data

– Is the necessary data available or can it be economically obtained?

• If the need is to forecast sales of a new product, then a customer survey may not be practical;

instead, historical analogy or market research may have to be used.

Criteria for Selecting a Forecasting Method

(78)

• Time Span

– What operations resource is being forecast and for what purpose?

– Short-term staffing needs might best be

forecast with moving average or exponential smoothing models.

– Long-term factory capacity needs might best be predicted with regression or executive-

committee consensus methods.

Criteria for Selecting a Forecasting Method

(79)

• Nature of Products and Services

– Is the product/service high cost or high volume?

– Where is the product/service in its life cycle?

– Does the product/service have seasonal demand fluctuations?

Criteria for Selecting a Forecasting Method

(80)

• Impulse Response and Noise Dampening

– An appropriate balance must be achieved between:

• How responsive we want the forecasting model to be to changes in the actual demand data

• Our desire to suppress undesirable chance variation or noise in the demand data

Criteria for Selecting a Forecasting Method

(81)

Reasons for Ineffective Forecasting

• Not involving a broad cross section of people

• Not recognizing that forecasting is integral to business planning

• Not forecasting the right things

• Not selecting an appropriate forecasting method

• Not tracking the accuracy of the forecasting models

• Not recognizing that forecasts will always be

wrong

(82)

Forecasting in Small Businesses and Start-Up Ventures

• Forecasting for these businesses can be difficult for the following reasons:

– Not enough personnel with the time to forecast

– Personnel lack the necessary skills to develop good forecasts

– These businesses are not data-rich environments – Forecasting for new products/services is always

difficult, even for the experienced forecaster

(83)

Sources of Forecasting Data and Help

• Government agencies at the local, regional, state, and federal levels

• Industry associations

• Consulting companies

(84)

Some Specific Forecasting Data

• Consumer Confidence Index

• Consumer Price Index (CPI)

• Gross Domestic Product (GDP)

• Index of Leading Economic Indicators

• Personal Income and Consumption

• Producer Price Index (PPI)

• Purchasing Manager’ s Index

• Retail Sales

(85)

Exponential Smoothing with Trend. As we move toward medium-range forecasts, trend becomes more important.