4-1 Operations Management Forecasting Chapter 4 - Part 2

4-1

Operations Operations ManagementManagement

ForecastingForecastingChapter 4 - Part 2Chapter 4 - Part 2

4-2

Trend is increasing or decreasing pattern.

First, plot data to verify trend.

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

Forecasting a TrendForecasting a Trend

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Plot DataPlot Data

Period

Actual

4 532 1 6

8

4

12

16

20

4-4

MA = 3 period Moving Average

Moving Averages for a TrendMoving Averages for a Trend

Period MA

1 8 2 11 3 13 4 15 10.67 4.33 5 19 13.00 6.00

MAError

6 15.67 ?

Sales

4-5

Trend Graph – Moving AverageTrend Graph – Moving Average

Period

Actual

MA Forecast

4 532 1 6

8

4

12

16

20

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

Exponential Smoothing for a Exponential Smoothing for a TrendTrend

?

Period ES

1 8 2 11 3 13 11 4 15 12 3.0 5 19 13.5 5.5

ESError

6

11

16.25

Sales

4-7

Trend Graph – Exponential Trend Graph – Exponential Smoothing and Moving AverageSmoothing and Moving Average

Period

Actual

MA Forecast

4 532 1 6

8

4

12

16

20ES Forecast

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

For a linear trend can use: Exponential Smoothing with Trend Adjustment

(pp. 115-117). Linear Trend Projection (linear regression).

For non-linear trend can use: Non-linear regression techniques.

Forecasting a TrendForecasting a Trend

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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.

Linear Trend ProjectionLinear Trend Projection

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Plot of X,Y DataPlot of X,Y Data

Time (x)

Valu

es o

f Dep

ende

nt V

aria

ble

(Y)

Actual observation

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Least SquaresLeast Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time (x)

Valu

es o

f Dep

ende

nt V

aria

ble

(Y)

bxaY ˆ

Actual observation

Point on regression line

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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.

4-13

Least Squares EquationsLeast Squares Equations

Equation: y = a + bx

Slope (p. 119):

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept: xbya

4-14

Linear Trend Projection ExampleLinear Trend Projection Example

5

Period

(x) 1 8 2 11 3 13 4 15

19

Sales(y)

Given the sales for last 5 periods, forecast future sales using trend projection.

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Linear Trend Projection ExampleLinear Trend Projection Example

5

4.55

156.2

5

666.2

3555

2.13352242

ab

Period

(x) 1 8 2 11 3 13 4 15

19

Sales(y) xy

6095

8 22 39

xy=224

x2

9 16 25

4 1

x2=55x=3 y=13.2

4-16

TP = Trend Projection: Y = 5.4 + 2.6x

Linear Trend Projection ExampleLinear Trend Projection ExamplePeriod

(x) MA ES

1 2 3 4 5

811131519

MAErr.

6

10.6713.0015.67

11 12

13.5

11

16.25

4.336.00

Sales(y)

3.05.5

ESErr.

TPErr.TP

21.018.415.8 -0.8

0.6

Small errors!

4-17

Trend GraphTrend Graph

MA Forecast

ES Forecast

Period

Actual

4 532 1 6

8

4

12

16

20

TP Forecast

4-18

Models with SeasonalityModels 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 120-124.

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Identify Independent and Dependent variables. 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.

Associative Forecasting Methods Associative Forecasting Methods

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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!

Independent variable (number of ads).

Y Xi i= +a b

Dependent variable (sales).

Linear RegressionLinear Regression

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Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Values of Independent Variable (x)

Valu

es o

f Dep

ende

nt V

aria

ble

(Y)

bxaY ˆ

Actual observation

Point on regression line

4-22

Linear Regression EquationsLinear Regression Equations(same as before)(same as before)

Equation: ii bxaY

Slope:

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept: xbya

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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.

Interpretation of CoefficientsInterpretation of Coefficients

4-24

Least SquaresLeast 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.

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Monthly Sales vs. Number of AdsMonthly Sales vs. Number of Ads

Number of TV ads per month

Sale

s

0

4-26

Least Squares LineLeast Squares Line


Sale

s

bxaY ˆ0

What is sales forecast for small number of ads?

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Forecasting Outside Range of Forecasting Outside Range of Observed Values is UnreliableObserved Values is Unreliable


Sale

s

bxaY ˆ0

Forecast is for negative sales!

4-28

Answers: ‘How strong is 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

CorrelationCorrelation

4-29

Sample Coefficient of CorrelationSample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

4-30

r = +1 r = -1

r = .89 r = 0

Y

X

Y

X

Y

XX

Coefficient of CorrelationCoefficient of Correlation

Y

4-31

A good forecast has: No pattern or direction in forecast error.

Error = Actual - Forecast

A small forecast error. Mean square error (MSE). Mean absolute deviation (MAD). Mean absolute percentage error (MAPE).

Guidelines for Selecting Guidelines for Selecting Forecasting ModelForecasting Model

4-32

Time

Error

0

Desired Pattern

Time

Error

0

Trend Not Fully Accounted for

Pattern of Forecast ErrorPattern of Forecast Error

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Suppose you have forecast sales with a linear regression model & exponential smoothing. Which model do you use?

Linear Regression Exponential Actual Model Smoothing

Year Sales Forecast Forecast (.9)1 1 0.6 1.002 1 1.3 1.003 2 2.0 1.004 2 2.7 1.905 4 3.4 1.99

Selecting Forecasting Model Selecting Forecasting Model ExampleExample

4-34

MSE = Σ Error2 / n = 1.10 / 5 = 0.220MAD = Σ |Error| / n = 2.0 / 5 = 0.400MAPE = Σ[|Error|/Actual]/n = 1.2/5 = 0.24 = 24%

Linear Regression ModelLinear Regression Model

1.10

Year Actual F’cast

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 0.6Total 0.0 2.0

Error Error2 |Error|

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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%

Exponential Smoothing ModelExponential Smoothing Model

Year Y i F’cast

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 2.01Total 0.3 5.05 3.11

Error Error2 |Error|

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Which is Better???Which is Better???

Linear Regression Model:MSE = Σ Error2 / n = 1.10 / 5 = 0.220MAD = Σ |Error| / n = 2.0 / 5 = 0.400MAPE = Σ[|Error|/Actual]/n = 1.2/5 = 0.24 = 24%

Exponential Smoothing Model:MSE = Σ Error2 / n = 5.05 / 5 = 1.01MAD = Σ |Error| / n = 3.11 / 5 = 0.622MAPE = Σ[|Error|/Actual]/n = 1.0525/5 = 0.2105 = 21%

4-37

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. Signal should be within upper and lower control

limits based on MAD.

Tracking SignalTracking Signal

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Plot of a Tracking SignalPlot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

4-39

Tracking Signal EquationTracking Signal Equation

MAD

error

MAD

yy

MAD

RSFETS

n

iii

1

ˆ

)(

4-40

Based on Normal Distribution of forecast errors: 1 MAD = approximately 0.8 standard deviations. Limits at ±3 MAD (±2.4 std. dev.) mean that 98% of

values should be within limits. Limits at ±4 MAD (±3.2 std. dev.) mean that 99.9% of

values should be within limits.

Use smaller limits to better control important items. (For example: ±2 MAD)

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

Tracking Signal LimitsTracking Signal Limits

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Tracking Signal - Month 1Tracking Signal - Month 1

MoMo F’cstF’cst ActAct ErrorError RSFERSFE MADMAD TSTS

11 100100 9090

CumCum|Error||Error|

4-42



11 100100 9090 -10-10 -10-10


RSFE = Errors = -10RSFE = Errors = -10

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


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11 100100 9090 -10-10 -10-10 1010


Cum |Error| = |Errors| = 10

Cum |Error| = |Errors| = 10

4-44



11 100100 9090 -10-10 -10-10 1010 10.010.0


MAD = |Errors|/n = 10/1 = 10

MAD = |Errors|/n = 10/1 = 10

4-45



11 100100 9090 -10-10 -10-10 1010 10.010.0 -1-1


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

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

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11 100100 9090

22 99 99 9494

-10-10 -10-10 1010 10.010.0 -1-1


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11 100100 9090

22 9999 9494

-10-10 -10-10 1010 10.010.0 -1-1

-5-5




4-48



11 100100 9090

22 9999 9494

-10-10 -10-10 1010 10.010.0 -1-1

-5-5 -15-15


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

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

4-49



11 100100 9090

22 9999 9494

-10-10 -10-10 1010 10.010.0 -1-1

-5-5 -15-15 1515


Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

4-50



11 100100 9090

22 9999 9494

-10-10 -10-10 1010 10.010.0 -1-1

-5-5 -15-15 1515 7.57.5


MAD = |Errors|/n = 15/2 = 7.5

MAD = |Errors|/n = 15/2 = 7.5

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11 100100 9090

22 9999 9494

-10-10 -10-10 1010 10.010.0 -1-1

-5-5 -15-15 1515 7.57.5 -2-2


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

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

4-52



11 100100 9090

22 9999 9494

33 9898 113113

-10-10 -10-10 1010 10.010.0 -1-1

-5-5 -15-15 1515 7.57.5 -2-2

1515 00 3300

1100

00


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Tracking Signal - Months 4-6Tracking Signal - Months 4-6


11 100100 9090

22 9999 9494

33 9898 113113

44 105105 9595

55 104104 119119

66 110110 140140

-10-10 -10-10 1010 10.010.0 -1-1

-5-5 -15-15 1515 7.57.5 -2-2

1515 00 3300

1100

00

-10-10 -10-10 4400

1010 -1-1

1515 55 5555

1111

.45.45

3030 3535 8855

14.214.2 2.472.47


4-54

Demand and ForecastDemand and Forecast

708090

100110120130140

0 1 2 3 4 5 6 7

Month

Forecast

Actual demand

4-55

Tracking SignalTracking Signal

0 1 2 3 4 5 6 7

Time

-3

-2

-1

0

1

2

3

Tra

ckin

g Si

gnal

98% of points should be between these limits.

4-56

Suppose you have forecast sales with a linear regression model & exponential smoothing. Which model do you use?

Linear Regression Exponential Actual Model Smoothing

Year Sales Forecast Forecast (.9)1 1 0.6 1.002 1 1.3 1.003 2 2.0 1.004 2 2.7 1.905 4 3.4 1.99

Selecting Forecasting Model Selecting Forecasting Model Example - RevisitedExample - Revisited

4-57

Linear Regression Model Linear Regression Model Tracking SignalTracking Signal

Year Y i F’cast

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 4 2 2.7 -0.7 0.35 -1.71 5 4 3.4 0.6 0.40 0.0

Error MAD TS

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Exponential Smoothing Model Exponential Smoothing Model Tracking SignalTracking Signal

1.99

Year Y i F’cast

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

Error MAD TS

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Tracking SignalsTracking Signals

50 1 2 3 4

Year

-3

-2

-1

0

1

2

3

Tra

ckin

g Si

gnal

Exponential Smoothing

Linear Regression

4-60

Forecasting in the Service SectorForecasting in the Service Sector

Examples: For staffing hospitals, fast-food restaurants, banking, etc.

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.

4-61

Forecasting SummaryForecasting Summary

Determine purpose of forecast first.

Plot data.

Use several appropriate methods.

Continually monitor, evaluate and adjust methods to improve forecasts.

Documents

4-1 Operations Management Forecasting Chapter 4 - Part 2