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# Operations Management Forecasting Chapter 4 - Part 2 - PowerPoint PPT Presentation

Operations Management Forecasting Chapter 4 - Part 2. Forecasting a Trend. Trend is increasing or decreasing pattern. First, plot data to verify trend. If trend exists, then moving averages and exponential smoothing will always lag. 20. Actual. 16. 12. 8. 4. 1. 4. 5. 2. 3. 6.

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Operations ManagementForecastingChapter 4 - Part 2

• Trend is increasing or decreasing pattern.

• First, plot data to verify trend.

• If trend exists, then moving averages and exponential smoothing will always lag.

Actual

16

12

8

4

1

4

5

2

3

6

Period

Plot Data

MA

Error

MA

Period

Sales

1

8

2

11

3

13

4.33

4

15

10.67

6.00

13.00

5

19

15.67

?

6

MA = 3 period Moving Average

20

Actual

16

MA Forecast

12

8

4

1

4

5

2

3

6

Period

MA

Error

ES

Error

ES

MA

Period

Sales

1

8

11

2

11

11

3

13

4.33

12

4

15

10.67

3.0

6.00

13.5

13.00

5

19

5.5

15.67

?

6

16.25

?

MA = 3 period Moving Average

ES = Exponential Smoothing with  =0.5 (F2=11)

ES Forecast

Actual

16

MA Forecast

12

8

4

1

4

5

2

3

6

Period

Trend Graph

• Moving Averages and (simple) Exponential Smoothing are always poor.

• For a linear trend can use:

• Exponential Smoothing with Trend Adjustment (skip: pp. 90-92).

• Linear Trend Projection (linear regression).

• For non-linear trend can use:

• Non-linear regression techniques.

• Used for forecasting linear trend line.

• PLOT TO VERIFY LINEAR RELATIONSHIP

• Assumes linear relationship between response variable, Y, and time, X.

• Y = a + bX

• a = y-axis intercept; b = slope

• Estimated by least squares method.

• Minimizes sum of squared errors.

Actual observation

Values of Dependent Variable (Y)

Time (x)

Actual observation

Deviation

Deviation

Deviation

Deviation

Values of Dependent Variable (Y)

Deviation

Point on regression line

Deviation

Deviation

Time (x)

• Least squares line minimizes sum of squared deviations.

• This reduces large errors.

• Similar to MSE.

• Deviations around least squares line are assumed to be random.

Slope (p. 94):

Y-Intercept:

Least Squares Equations

(x)

Sales

(y)

x2

xy

1

1

8

8

4

2

11

22

39

9

3

13

60

16

4

15

25

5

19

95

x=3

xy=224

x2=55

y=13.2

Linear Trend Projection Example

(x)

TP

Err.

Sales

(y)

ES

Err.

MA

Err.

ES

MA

TP

1

8

11

2

11

11

3

13

12

4

15

10.67

15.8

3.0

4.33

-0.8

13.5

18.4

13.00

5

19

0.6

5.5

6.00

15.67

6

16.25

21.0

Linear Trend Projection Example

TP = Trend Projection: Y = 5.4 + 2.6x

Small errors!

Actual

16

12

8

4

1

4

5

2

3

6

Period

Trend Graph

TP Forecast

ES Forecast

MA Forecast

• Use if data exhibits seasonal patterns.

• Daily, weekly, monthly, yearly.

• Compute seasonal component.

• Remove seasonality and forecast.

• Factor in seasonal component.

• See pages 96-100.

• Identify Independent and dependent variable.

• Dependent variable (y): Entity to be forecast (demand).

• Independent variable (x): Used to predict (or explain) dependent variable.

• Determine relationship.

• Plot data.

• Consider time lags.

• Calculate parameters.

• Forecast.

• Monitor.

• Linear relationship between dependent & explanatory variables.

• Example: Sales in month i (Yi ) depends on advertising in month i (Xi ) (eg. number of ads)

• Sales may also depend on advertising in previous months!

Y

a

+

b

X

=

i

i

Dependent variable (sales).

Actual observation

Deviation

Deviation

Deviation

Deviation

Values of Dependent Variable (Y)

Deviation

Point on regression line

Deviation

Deviation

Values of Independent Variable (x)

Y-Intercept:

Linear Regression Equations(same as before)

Slope:

• Slope (b):

• Y changes by b units for each 1 unit increase in X.

• If b = +2, then sales (Y) is forecast to increase by 2 for each 1 unit increase in advertising (X).

• Y-intercept (a):

• Average value of Y when X = 0.

• If a = 4, then average sales (Y) is expected to be 4 when advertising (X) is 0.

• Plot data to verify linearity!

• If curve is present, use non-linear regression.

• Forecast only in (or near) range of observed values!

• May need future values of independent variable to make forecast.

• Example: Summer hotel demand may depend on summer gasoline price.

Sales

0

Number of TV ads per month

What is sales forecast for small number of ads?

Sales

0

Number of TV ads per month

Forecast is for negative sales!

Sales

0

Number of TV ads per month

• Answers: ‘How strongis the linear relationship between the variables?’

• Coefficient of correlation - r

• Measures degree of association; ranges from -1 to +1

• Coefficient of determination - r2

• Amount of variation explained by regression equation.

• Used to evaluate quality of linear relationship.

r = -1

Y

Y

X

X

r = .89

r = 0

Y

X

X

Coefficient of Correlation

Y

• You want to achieve:

• No pattern or direction in forecast error.

• Error = Actual - Forecast

• Small forecast error.

• Mean square error (MSE).

• Mean absolute percentage error (MAPE).

Trend Not Fully Accounted for

Error

Error

0

0

Time

Time

Pattern of Forecast Error

You’re a marketing analyst for Hasbro Toys. You’ve forecast sales with a linear regression model & exponential smoothing. Which model do you use?

Linear Regression Exponential

Actual Model Smoothing

YearSales Forecast Forecast (.9)

1 1 0.6 1.00 2 1 1.3 1.00 3 2 2.0 1.00 4 2 2.7 1.90 5 4 3.4 1.99

F’cast

Error

Error2

|Error|

Year

i

1

1

0.6

0.4

0.16

0.4

2

1

1.3

-0.3

0.09

0.3

3

2

2.0

0.0

0.00

0.0

4

2

2.7

-0.7

0.49

0.7

5

4

3.4

0.6

0.36

0.6

Total

0.0

2.0

Linear Regression Model

1.10

MSE = Σ Error2 / n = 1.10 / 5 = 0.220

MAD = Σ |Error| / n = 2.0 / 5 = 0.400

MAPE = Σ[|Error|/Actual]/n = 1.2/5 = 0.24 = 24%

Year

F’cast

Error

Error2

|Error|

i

1

1

1.00

0.0

0.00

0.0

2

1

1.00

0.0

0.00

0.0

3

2

1.00

1.0

1.00

1.0

4

2

1.90

0.1

0.01

0.1

5

4

2.01

4.04

2.01

Total

0.3

5.05

3.11

Exponential Smoothing Model

1.99

MSE = Σ Error2 / n = 5.05 / 5 = 1.01

MAD = Σ |Error| / n = 3.11 / 5 = 0.622

MAPE = Σ[|Error|/Actual]/n = 1.0525/5 = 0.2105 = 21%

Linear Regression Model:

MSE = Σ Error2 / n = 1.10 / 5 = 0.220

MAD = Σ |Error| / n = 2.0 / 5 = 0.400

MAPE = Σ[|Error|/Actual]/n = 1.2/5 = 0.24 = 24%

Exponential Smoothing Model:

MSE = Σ Error2 / n = 5.05 / 5 = 1.01

MAD = Σ |Error| / n = 3.11 / 5 = 0.622

MAPE = Σ[|Error|/Actual]/n = 1.0525/5 = 0.2105 = 21%

• Measures how well the forecast is predicting actual values.

• To use:

• Calculate tracking signal each time period.

• Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD).

• Plot tracking signal on graph.

• Good tracking signal has low values.

• Should be within upper and lower control limits (often based on MAD).

Signal exceeded limit

Tracking signal

Upper control limit

+

0

Acceptable range

-

Lower control limit

Time

Mo

F’cst

Act

RSFE

TS

Error

|Error|

1

100

90

Tracking Signal - Month 1

Mo

F’cst

Act

RSFE

TS

Error

|Error|

-10

-10

1

100

90

Error = Actual - Forecast = 90 - 100 = -10

RSFE =  Errors = -10

Tracking Signal - Month 1

Mo

F’cst

Act

RSFE

TS

Error

|Error|

-10

10

-10

1

100

90

Cum |Error| =  |Errors| = 10

Tracking Signal - Month 1

Mo

F’cst

Act

RSFE

TS

Error

|Error|

-10

10

-10

1

100

90

10.0

MAD =  |Errors|/n = 10/1 = 10

Tracking Signal - Month 1

Mo

F’cst

Act

RSFE

TS

Error

|Error|

-10

10

-10

1

100

90

10.0

-1

TS = RSFE/MAD = -10/10 = -1

Tracking Signal - Month 1

Mo

F’cst

Act

RSFE

TS

Error

|Error|

-10

10

-10

1

100

90

10.0

-1

2

99

94

Tracking Signal - Month 2

Mo

F’cst

Act

RSFE

TS

Error

|Error|

-10

10

-10

1

100

90

10.0

-1

-5

2

99

94

Error = Actual - Forecast = 94 - 99 = -5

Tracking Signal - Month 2

Mo

F’cst

Act

RSFE

TS

Error

|Error|

-10

10

-10

1

100

90

10.0

-1

-15

-5

2

99

94

RSFE =  Errors = (-10) + (-5) = -15

Tracking Signal - Month 2

Mo

F’cst

Act

RSFE

TS

Error

|Error|

-10

10

-10

1

100

90

10.0

-1

-15

15

-5

2

99

94

Cum Error =  |Errors| = 10 + 5 = 15

Tracking Signal - Month 2

Mo

F’cst

Act

RSFE

TS

Error

|Error|

-10

10

-10

1

100

90

10.0

-1

-15

15

-5

2

99

94

7.5

MAD =  |Errors|/n = 15/2 = 7.5

Tracking Signal - Month 2

Mo

F’cst

Act

RSFE

TS

Error

|Error|

-10

10

-10

1

100

90

10.0

-1

-15

15

-5

2

99

94

7.5

-2

TS = RSFE/MAD = -15/7.5 = -2

Tracking Signal - Month 2

Mo

F’cst

Act

RSFE

TS

Error

|Error|

-10

10

-10

1

100

90

10.0

-1

-15

15

-5

2

99

94

7.5

-2

30

10

0

15

0

3

98

113

Tracking Signal - Month 3

Mo

F’cst

Act

RSFE

TS

Error

|Error|

-10

10

-10

1

100

90

10.0

-1

-15

15

-5

2

99

94

7.5

-2

30

10

0

15

0

3

98

113

-10

40

-10

10

-1

4

105

95

5

15

55

11

.45

5

104

119

14.2

35

30

85

2.47

6

110

140

Tracking Signal - Months 4-6

140

Forecast

130

120

110

100

90

80

Actual demand

70

0

1

2

3

4

5

6

7

Month

3

2

1

Tracking Signal

0

-1

-2

-3

0

1

2

3

4

5

6

7

Time

• Upper and lower limits depend on the product being forecast.

• 98% of values should be within  3 MAD.

• 99.9% of values should be within  4 MAD.

• Use smaller limits for high volume items.

• Patterns, even if within limits, indicate better forecasts can be made.

You’re a marketing analyst for Hasbro Toys. You’ve forecast sales with a linear regression model & exponential smoothing. Which model do you use?

Linear Regression Exponential

Actual Model Smoothing

YearSales Forecast Forecast (.9)

1 1 0.6 1.00 2 1 1.3 1.00 3 2 2.0 1.00 4 2 2.7 1.90 5 4 3.4 1.99

Y

F’cast

Error

Year

i

1

1

0.6

0.4

0.4

1.0

2

1

1.3

-0.3

0.35

0.29

3

2

2.0

0.0

0.233

0.43

-1.71

4

2

2.7

-0.7

0.35

5

4

3.4

0.6

0.40

0.0

Linear Regression Model Tracking Signal

Year

F’cast

Error

TS

i

1

1

1.00

0.0

0.0

0.0

2

1

1.00

0.0

0.0

0.0

3

2

1.00

1.0

0.33

3.0

4

2

1.90

0.1

0.275

4.0

5

4

2.01

0.622

5.0

1.99

Exponential Smoothing Model Tracking Signal

Exponential Smoothing

3

2

1

0

TrackingSignal

-1

Linear Regression

-2

-3

0

1

2

3

4

5

Year

• Presents unusual challenges:

• Large variability (during day, week, etc.).

• Special need for short term forecasting.

• Needs differ greatly as function of industry and product.

• Issues of holidays and calendar.

• Examples: Staffing for hospitals, fast-food restaurants, banking, etc.

• Determine purpose of forecast first.

• Plot data.

• Use several appropriate methods.

• Continually monitor, evaluate and adjust methods to improve forecasts.