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CHAPTER 13 FORECASTING. Outline Forecasting and Choice of a Forecasting Methods Methods for Stationary Series: Simple and Weighted Moving Average Exponential smoothing Trend-Based Methods Regression Double Exponential Smoothing: Holt’s Method A Method for Seasonality and Trend.

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CHAPTER 13FORECASTING

Outline

  • Forecasting and Choice of a Forecasting Methods

  • Methods for Stationary Series:

    • Simple and Weighted Moving Average

    • Exponential smoothing

  • Trend-Based Methods

    • Regression

    • Double Exponential Smoothing: Holt’s Method

  • A Method for Seasonality and Trend



Production

Aggregate planning, inventory control, scheduling

Marketing

New product introduction, sales-force allocation, promotions

Finance

Plant/equipment investment, budgetary planning

Personnel

Workforce planning, hiring, layoff

Decisions Based on Forecasts


Characteristics of forecasts

Forecasts are always wrong; so consider both expected value and a measure of forecast error

Long-term forecasts are less accurate than short-term forecasts

Aggregate forecasts are more accurate than disaggregate forecasts

Characteristics of Forecasts


Forecasting and a measure of forecast error

  • Components of demand

  • Evaluation of forecasts

  • Time series: stationary series

  • Time series: trend

    • Linear regression

    • Double exponential smoothing

  • Time series: seasonality


Components of Demand and a measure of forecast error

  • Average demand

  • Trend

    • Gradual shift in average demand

  • Seasonal pattern

    • Periodic oscillation in demand which repeats

  • Cycle

    • Similar to seasonal patterns, length and magnitude of the cycle may vary

  • Random movements

  • Auto-correlation


Components of Demand and a measure of forecast error

Qantity

Time

(a) Average: Data cluster about a horizontal line.


Components of Demand and a measure of forecast error

Quantity

Time

(b) Linear trend: Data consistently increase or decrease.


Components of Demand and a measure of forecast error

Year 1

Quantity

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

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

Months

(c) Seasonal influence: Data consistently show peaks and valleys.


Components of Demand and a measure of forecast error

Year 1

Quantity

Year 2

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

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

Months

(c) Seasonal influence: Data consistently show peaks and valleys.


Components of Demand and a measure of forecast error


Components of Demand and a measure of forecast error

Quantity

| | | | | |

1 2 3 4 5 6

Years

(c) Cyclical movements: Gradual changes over extended periods of time.


Components of Demand and a measure of forecast error


Components of Demand and a measure of forecast error

Trend

Demand

Random

movement

Time

Demand

Trend with

seasonal pattern

Time


Snow Skiing and a measure of forecast error

Seasonal

Long term growth trend

Demand for skiing products increased sharply after the Nagano Olympics


Choosing a method forecast error

Measures of Forecast Error and a measure of forecast error

Et = At - Ft

RSFE = Et

MAD =

MSE =

MAPE =

 = MSE

|Et |

n

Et2

n

[|Et | (100)]/At

n

Choosing a MethodForecast Error


Choosing a method forecast error1

Absolute and a measure of forecast error

Error Absolute Percent

Month, Demand, Forecast, Error, Squared, Error, Error,

tAtFtEt Et2 |Et| (|Et|/At)(100)

1 200 225

2 240 220

3 300 285

4 270 290

5 230 250

6 260 240

7 210 250

8 275 240

-

Total

Choosing a MethodForecast Error


Choosing a method forecast error2

MSE = = and a measure of forecast error

MAD = =

MAPE = =

Choosing a MethodForecast Error

Measures of Error

RSFE =


Choosing a method forecast error3

Running Sum Mean Absolute and a measure of forecast error

of Forecast Errors Deviation

Method (RSFE - bias) (MAD)

Simple moving average

Three-week (n = 3) 23.1 17.1

Six-week (n = 6) 69.8 15.5

Weighted moving average

0.70, 0.20, 0.10 14.0 18.4

Exponential smoothing

 = 0.1 65.6 14.8

 = 0.2 41.0 15.3

Choosing a MethodForecast Error


Choosing a method tracking signals

RSFE and a measure of forecast error

MAD

Tracking signal =

+2.0 —

+1.5 —

+1.0 —

+0.5 —

0 —

- 0.5 —

- 1.0 —

- 1.5 —

Control limit

Tracking signal

Control limit

| | | | |

0 5 10 15 20 25

Observation number

Choosing a MethodTracking Signals


Choosing a method tracking signals1

RSFE and a measure of forecast error

MAD

Tracking signal =

+2.0 —

+1.5 —

+1.0 —

+0.5 —

0 —

- 0.5 —

- 1.0 —

- 1.5 —

Out of control

Control limit

Tracking signal

Control limit

| | | | |

0 5 10 15 20 25

Observation number

Choosing a MethodTracking Signals


Choosing a Method and a measure of forecast errorTracking Signals


Problem 13-2: and a measure of forecast errorHistorical demand for a product is:

Month Jan Feb Mar Apr May Jun

Demand 12 11 15 12 16 15

a. Using a weighted moving average with weights of 0.60, 0.30, and 0.10, find the July forecast.

b. Using a simple three-month moving average, find the July forecast.

c. Using single exponential smoothing with =0.20 and a June forecast =13, find the July forecast.

d. Using simple regression analysis, calculate the regression equation for the preceding demand data

e. Using regression equation in d, calculate the forecast in July


Problem 13-15: and a measure of forecast errorIn this problem, you are to test the validity of your forecasting model. Here are the forecasts for a model you have been using and the actual demands that occurred:

Week 1 2 3 4 5 6

Forecast 800 850 950 950 1,000 975

Actual 900 1,000 1,050 900 900 1,100

Compute MAD and tracking signal. Then decide whether the forecasting model you have been using is giving reasonable results.


Methods for Stationary Series and a measure of forecast error


Time series methods simple moving averages

450 — and a measure of forecast error

430 —

410 —

390 —

370 —

Patient arrivals

Actual patient

arrivals

| | | | | |

0 5 10 15 20 25 30

Time Series MethodsSimple Moving Averages

Week


Time series methods simple moving averages1

Time Series Methods and a measure of forecast errorSimple Moving Averages

450 —

430 —

410 —

390 —

370 —

Patient

Week Arrivals

1 400

2 380

3 411

Patient arrivals

Actual patient

arrivals

| | | | | |

0 5 10 15 20 25 30

Week


Time series methods simple moving averages2

Time Series Methods and a measure of forecast errorSimple Moving Averages

450 —

430 —

410 —

390 —

370 —

Patient

Week Arrivals

1 400

2 380

3 411

Patient arrivals

F4 =

Actual patient

arrivals

| | | | | |

0 5 10 15 20 25 30

Week


Time series methods simple moving averages3

Time Series Methods and a measure of forecast errorSimple Moving Averages

450 —

430 —

410 —

390 —

370 —

Patient

Week Arrivals

2 380

3 411

4 415

Patient arrivals

F5 =

Actual patient

arrivals

| | | | | |

0 5 10 15 20 25 30

Week


Time series methods simple moving averages4

450 — and a measure of forecast error

430 —

410 —

390 —

370 —

3-week MA

forecast

Patient arrivals

Actual patient

arrivals

| | | | | |

0 5 10 15 20 25 30

Week

Time Series MethodsSimple Moving Averages


Time series methods simple moving averages5

450 — and a measure of forecast error

430 —

410 —

390 —

370 —

6-week MA

forecast

3-week MA

forecast

Patient arrivals

Actual patient

arrivals

| | | | | |

0 5 10 15 20 25 30

Time Series MethodsSimple Moving Averages

Week


Taco Bell determined that the demand for each 15-minute interval

can be estimated from a 6-week simple moving average of sales.

The forecast was used to determine the number of employees needed.


Time series methods weighted moving average

Assigned weights interval

t-1 0.70

t-2 0.20

t-3 0.10

Time Series MethodsWeighted Moving Average

450 —

430 —

410 —

390 —

370 —

3-week MA

forecast

Weighted Moving Average

Patient arrivals

F4 =

Actual patient

arrivals

| | | | | |

0 5 10 15 20 25 30

Week


Time series methods weighted moving average1

Assigned weights interval

t-1 0.70

t-2 0.20

t-3 0.10

Time Series MethodsWeighted Moving Average

450 —

430 —

410 —

390 —

370 —

3-week MA

forecast

Weighted Moving Average

Patient arrivals

F5 =

Actual patient

arrivals

| | | | | |

0 5 10 15 20 25 30

Week


Time series methods exponential smoothing

Time Series Methods interval Exponential Smoothing

450 —

430 —

410 —

390 —

370 —

Exponential Smoothing

 = 0.10

Ft = At-1 + (1 - )Ft - 1

Patient arrivals

| | | | | |

0 5 10 15 20 25 30

Week


Time series methods exponential smoothing1

Time Series Methods interval Exponential Smoothing

450 —

430 —

410 —

390 —

370 —

Exponential Smoothing

 = 0.10

Ft = At-1 + (1 - )Ft - 1

F3 = (400 + 380)/2=390

A3 = 411

Patient arrivals

| | | | | |

0 5 10 15 20 25 30

Week


Time series methods exponential smoothing2

Time Series Methods interval Exponential Smoothing

450 —

430 —

410 —

390 —

370 —

Exponential Smoothing

 = 0.10

Ft = At-1 + (1 - )Ft - 1

F3 = (400 + 380)/2=390

A3 = 411

Patient arrivals

F4 =

| | | | | |

0 5 10 15 20 25 30

Week


Time series methods exponential smoothing3

Time Series Methods interval Exponential Smoothing

450 —

430 —

410 —

390 —

370 —

Exponential Smoothing

 = 0.10

Ft = At + (1 - )Ft - 1

F4 =

A4 = 415

Patient arrivals

F5 =

| | | | | |

0 5 10 15 20 25 30

Week


Time series methods exponential smoothing4

450 — interval

430 —

410 —

390 —

370 —

Patient arrivals

| | | | | |

0 5 10 15 20 25 30

Week

Time Series MethodsExponential Smoothing


Comparison of exponential smoothing and simple moving average
Comparison of Exponential Smoothing and Simple Moving Average

  • Both Methods

    • Are designed for stationary demand

    • Require a single parameter

    • Lag behind a trend, if one exists

    • Have the same distribution of forecast error if


Comparison of Exponential Smoothing and Simple Moving Average

  • Moving average uses only the last N periods data, exponential smoothing uses all data

  • Exponential smoothing uses less memory and requires fewer steps of computation; store only the most recent forecast!


Problem 13-20: AverageYour manager is trying to determine what forecasting method to use. Based upon the following historical data, calculate the following forecast and specify what procedure you would utilize:

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

Actual demand 62 65 67 68 71 73 76 78 78 80 84 85

a. Calculate the three-month SMA forecast for periods 4-12

b. Calculate the weighted three-month MA using weights of 0.50, 0.30, and 0.20 for periods 4-12.

c. Calculate the single exponential smoothing forecast for periods 2-12 using an initial forecast, F1=61 and =0.30

d. Calculate the exponential smoothing with trend component forecast for periods 2-12 using T1=1.8,F1=60,=0.30,=0.30

e. Calculate MAD for the forecasts made by each technique in periods 4-12. Which forecasting method do you prefer?



Turkeys have a long-term trend for increasing demand with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January.


Linear regression

Y with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January.

Dependent variable

X

Independent variable

Linear Regression


Linear regression1

Regression with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January.

equation:

Y = a + bX

Y

Dependent variable

X

Independent variable

Linear Regression


Linear regression2

Regression with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January.

equation:

Y = a + bX

Y

Actual

value

of Y

Dependent variable

Value of X used

to estimate Y

X

Independent variable

Linear Regression


Linear regression3

Regression with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January.

equation:

Y = a + bX

Y

Estimate of

Y from

regression

equation

Actual

value

of Y

Dependent variable

Value of X used

to estimate Y

X

Independent variable

Linear Regression


Linear regression4

Deviation, with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January.

or error

Regression

equation:

Y = a + bX

Y

Estimate of

Y from

regression

equation

{

Actual

value

of Y

Dependent variable

Value of X used

to estimate Y

X

Independent variable

Linear Regression


Linear regression5

Sales Advertising with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January.

Month (000 units) (000 $)

1 264 2.5

2 116 1.3

3 165 1.4

4 101 1.0

5 209 2.0

Linear Regression


Linear regression6

Sales, y Advertising, x with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January.

Month (000 units) (000 $)

1 264 2.5

2 116 1.3

3 165 1.4

4 101 1.0

5 209 2.0

xy - nxy

x2 - n(x)2

a = y - bx

b =

Linear Regression


Linear regression7

Sales, with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January. y Advertising, x

Month (000 units) (000 $) xyx 2

1 264 2.5

2 116 1.3

3 165 1.4

4 101 1.0

5 209 2.0

Total

y= x =

xy - nxy

x 2 - nx 2

a = y - bx

b =

Linear Regression


300 — with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January.

250 —

200 —

150 —

100 —

50

Sales (000s)

| | | |

1.0 1.5 2.0 2.5

b = 109.229

Y =


Linear regression8

Sales, with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January. y Advertising, x

Month (000 units) (000 $) xyx 2y 2

1 264 2.5 660.0 6.25

2 116 1.3 150.8 1.69

3 165 1.4 231.0 1.96

4 101 1.0 101.0 1.00

5 209 2.0 418.0 4.00

Total 855 8.2 1560.8 14.90

y = 171 x = 1.64

nxy - x y

[nx 2 -(x) 2][ny 2 - (y) 2]

r =

Linear Regression


Sales, with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January. y Advertising, x

Month (000 units) (000 $) xyx 2y 2

1 264 2.5 660.0 6.25 69,696

2 116 1.3 150.8 1.69 13,456

3 165 1.4 231.0 1.96 27,225

4 101 1.0 101.0 1.00 10,201

5 209 2.0 418.0 4.00 43,681

Total 855 8.2 1560.8 14.90 164,259

y= 171 x = 1.64

r = 0.98 r 2 = 0.96

Linear Regression


Linear regression9

Linear Regression with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January.

Forecast for Month 6:

Advertising expenditure = $1750

Y =


Time series methods linear regression analysis

Time Series Methods with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January. Linear Regression Analysis

80 —

70 —

60 —

50 —

40 —

30 —

Yn = a + bXn

where

Xn = Weekn

Patient arrivals

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

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Week


Time series methods linear regression analysis1

Time Series Methods with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January. Linear Regression Analysis

80 —

70 —

60 —

50 —

40 —

30 —

Yn = a + bXn

where

Xn = Weekn

Patient arrivals

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

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Week


Time Series Methods with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January. Linear Regression Analysis

  • Standard error of estimate is computed as follows:


Time Series Methods with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January. Linear Regression Analysis

  • An use of the standard error of estimate:

    • Suppose that a manager forecasts that the demand for a product is 500 units and Syx is 20. If the manager wants to accept a stockout only 2% time, how many additional units should be held in the inventory?


Time Series Methods with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January. Double Exponential Smoothing

  • The method uses two smoothing constants  and 


A Comparison of Methods with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January.

90

85

Actual

3-Mo MA

80

3-Mo WMA

Demand

75

Exp Sm

70

Double Exp Sm

65

60

0

5

10

15

Months


Methods for seasonal series

Methods for Seasonal Series with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January.


Time series methods seasonal influences

Quarter Year 1 Year 2 Year 3 Year 4 with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January.

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

Time Series MethodsSeasonal Influences


Time series methods seasonal influences1

Quarter Year 1 Year 2 Year 3 Year 4 with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January.

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

Actual Demand

Average Demand

Seasonal Index =

Time Series MethodsSeasonal Influences


Time series methods seasonal influences2

Quarter Year 1 Year 2 Year 3 Year 4 with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January.

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 = =

Time Series MethodsSeasonal Influences


Time series methods seasonal influences3

Quarter Year 1 Year 2 Year 3 Year 4 with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January.

1 45/250 = 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 = =

Time Series MethodsSeasonal Influences


Time series methods seasonal influences4

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

Time Series MethodsSeasonal Influences


Time series methods seasonal influences5

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

Quarter Average Seasonal Index

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

2

3

4

Time Series MethodsSeasonal Influences


Time series methods seasonal influences6

Projected Annual Demand = 2600 Year 3 Year 4

Average Quarterly Demand = 2600/4 = 650

Quarter Average Seasonal Index Forecast

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

2

3

4

Time Series MethodsSeasonal Influences


Seasonal influences

(a) Multiplicative influence Year 3 Year 4

Demand

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

0 2 4 5 8 10 12 14 16

Period

Seasonal Influences


Seasonal influences1

(b) Additive influence Year 3 Year 4

Demand

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

0 2 4 5 8 10 12 14 16

Period

Seasonal Influences


Time Series Methods Year 3 Year 4Seasonal Influences with Trend

Step 1: Determine seasonal factors

  • Example: if the demands are quarterly, divide the average demand in Quarter 1 by the average quarterly demand

    Step 2: Deseasonalize the original data

  • Divide the original data by the seasonal factors

    Step 3: Develop a regression line on deaseasonalized data

  • Find parameters a and b in Y=a+bX

  • Where

  • yi = deseasonalized data (not the original data)

  • xi = time; 1, 2, 3, …, n

  • n = Number of periods


Time Series Methods Year 3 Year 4Seasonal Influences with Trend

Step 4: Make projection using regression line

  • For each i = n+1, n+2, …, compute yi by substituting a, b and xi in the regression equation yi= a+bxi

    Step 5: Reseasonalize projection using seasonal factors

  • Multiply the projected values by the seasonal factors


Problem 13-21: Year 3 Year 4 Use regression analysis on deseasonalized demand to forecast demand in summer 2006, given the following historical demand data:

Year Season Actual Demand

2004 Spring 205

Summer 140

Fall 375

Winter 575

2005 Spring 475

Summer 275

Fall 685

Winter 965


Reading and Exercises Year 3 Year 4

  • Chapter 13 pp. 518-539

  • Problems 1, 7, 13, 14,16


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