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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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forecasting a trend
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.
plot data

20

Actual

16

12

8

4

1

4

5

2

3

6

Period

Plot Data
moving averages for a trend
Moving Averages for a Trend

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

trend graph
Trend Graph

20

Actual

16

MA Forecast

12

8

4

1

4

5

2

3

6

Period

exponential smoothing for a trend
Exponential Smoothing for a Trend

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)

trend graph1

20

ES Forecast

Actual

16

MA Forecast

12

8

4

1

4

5

2

3

6

Period

Trend Graph
forecasting a trend1
Forecasting a Trend
  • 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.
linear trend projection
Linear Trend Projection
  • 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.
plot of x y data
Plot of X,Y Data

Actual observation

Values of Dependent Variable (Y)

Time (x)

least squares
Least Squares

Actual observation

Deviation

Deviation

Deviation

Deviation

Values of Dependent Variable (Y)

Deviation

Point on regression line

Deviation

Deviation

Time (x)

least squares1
Least Squares
  • 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.
least squares equations

Equation:

Slope (p. 94):

Y-Intercept:

Least Squares Equations
linear trend projection example

Period

(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
linear trend projection example1

Period

(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!

trend graph2

20

Actual

16

12

8

4

1

4

5

2

3

6

Period

Trend Graph

TP Forecast

ES Forecast

MA Forecast

models with seasonality
Models with Seasonality
  • 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.
associative forecasting methods
Associative Forecasting Methods
  • 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 regression
Linear Regression
  • 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

Independent variable (number of ads).

Dependent variable (sales).

least squares2
Least Squares

Actual observation

Deviation

Deviation

Deviation

Deviation

Values of Dependent Variable (Y)

Deviation

Point on regression line

Deviation

Deviation

Values of Independent Variable (x)

interpretation of coefficients
Interpretation of Coefficients
  • 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.
least squares3
Least Squares
  • 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.
monthly sales vs number of ads
Monthly Sales vs. Number of Ads

Sales

0

Number of TV ads per month

least squares line
Least Squares Line

What is sales forecast for small number of ads?

Sales

0

Number of TV ads per month

forecasting outside range of observed values is unreliable
Forecasting Outside Range of Observed Values is Unreliable

Forecast is for negative sales!

Sales

0

Number of TV ads per month

correlation
Correlation
  • 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.
coefficient of correlation

r = +1

r = -1

Y

Y

X

X

r = .89

r = 0

Y

X

X

Coefficient of Correlation

Y

guidelines for selecting forecasting model
Guidelines for Selecting Forecasting Model
  • You want to achieve:
    • No pattern or direction in forecast error.
      • Error = Actual - Forecast
    • Small forecast error.
      • Mean square error (MSE).
      • Mean absolute deviation (MAD).
      • Mean absolute percentage error (MAPE).
pattern of forecast error

Desired Pattern

Trend Not Fully Accounted for

Error

Error

0

0

Time

Time

Pattern of Forecast Error
selecting forecasting model example
Selecting Forecasting Model Example

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

linear regression model

Y

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%

exponential smoothing model

Y

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%

which is better
Which is Better???

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%

tracking signal
Tracking Signal
  • 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).
plot of a tracking signal
Plot of a Tracking Signal

Signal exceeded limit

Tracking signal

Upper control limit

+

0

MAD

Acceptable range

-

Lower control limit

Time

tracking signal month 1

Cum

Mo

F’cst

Act

RSFE

MAD

TS

Error

|Error|

1

100

90

Tracking Signal - Month 1
tracking signal month 11

Cum

Mo

F’cst

Act

RSFE

MAD

TS

Error

|Error|

-10

-10

1

100

90

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

RSFE =  Errors = -10

Tracking Signal - Month 1
tracking signal month 12

Cum

Mo

F’cst

Act

RSFE

MAD

TS

Error

|Error|

-10

10

-10

1

100

90

Cum |Error| =  |Errors| = 10

Tracking Signal - Month 1
tracking signal month 13

Cum

Mo

F’cst

Act

RSFE

MAD

TS

Error

|Error|

-10

10

-10

1

100

90

10.0

MAD =  |Errors|/n = 10/1 = 10

Tracking Signal - Month 1
tracking signal month 14

Cum

Mo

F’cst

Act

RSFE

MAD

TS

Error

|Error|

-10

10

-10

1

100

90

10.0

-1

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

Tracking Signal - Month 1
tracking signal month 2

Cum

Mo

F’cst

Act

RSFE

MAD

TS

Error

|Error|

-10

10

-10

1

100

90

10.0

-1

2

99

94

Tracking Signal - Month 2
tracking signal month 21

Cum

Mo

F’cst

Act

RSFE

MAD

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
tracking signal month 22

Cum

Mo

F’cst

Act

RSFE

MAD

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
tracking signal month 23

Cum

Mo

F’cst

Act

RSFE

MAD

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
tracking signal month 24

Cum

Mo

F’cst

Act

RSFE

MAD

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
tracking signal month 25

Cum

Mo

F’cst

Act

RSFE

MAD

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
tracking signal month 3

Cum

Mo

F’cst

Act

RSFE

MAD

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
tracking signal months 4 6

Cum

Mo

F’cst

Act

RSFE

MAD

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
demand and forecast
Demand and Forecast

140

Forecast

130

120

110

100

90

80

Actual demand

70

0

1

2

3

4

5

6

7

Month

tracking signal1
Tracking Signal

3

2

1

Tracking Signal

0

-1

-2

-3

0

1

2

3

4

5

6

7

Time

tracking signal limits
Tracking Signal Limits
  • 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.
    • For example: +2 MAD, -2 MAD
  • Patterns, even if within limits, indicate better forecasts can be made.
selecting forecasting model example revisited
Selecting Forecasting Model Example - Revisited

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

linear regression model tracking signal

TS

Y

F’cast

Error

MAD

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
exponential smoothing model tracking signal

Y

Year

F’cast

Error

TS

MAD

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
tracking signals
Tracking Signals

Exponential Smoothing

3

2

1

0

TrackingSignal

-1

Linear Regression

-2

-3

0

1

2

3

4

5

Year

forecasting in the service sector
Forecasting in the Service Sector
  • 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.
forecasting summary
Forecasting Summary
  • Determine purpose of forecast first.
  • Plot data.
  • Use several appropriate methods.
  • Continually monitor, evaluate and adjust methods to improve forecasts.
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