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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 13 FORECASTING

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Chapter 13 forecasting

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


Chapter 13 forecasting

Forecasting


Chapter 13 forecasting

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


Chapter 13 forecasting

Forecasting

  • Components of demand

  • Evaluation of forecasts

  • Time series: stationary series

  • Time series: trend

    • Linear regression

    • Double exponential smoothing

  • Time series: seasonality


Chapter 13 forecasting

Components of Demand

  • 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


Chapter 13 forecasting

Components of Demand

Qantity

Time

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


Chapter 13 forecasting

Components of Demand

Quantity

Time

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


Chapter 13 forecasting

Components of Demand

Year 1

Quantity

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

JFMAMJJASOND

Months

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


Chapter 13 forecasting

Components of Demand

Year 1

Quantity

Year 2

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

JFMAMJJASOND

Months

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


Chapter 13 forecasting

Components of Demand


Chapter 13 forecasting

Components of Demand

Quantity

||||||

123456

Years

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


Chapter 13 forecasting

Components of Demand


Chapter 13 forecasting

Components of Demand

Trend

Demand

Random

movement

Time

Demand

Trend with

seasonal pattern

Time


Chapter 13 forecasting

Snow Skiing

Seasonal

Long term growth trend

Demand for skiing products increased sharply after the Nagano Olympics


Choosing a method forecast error

Measures 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

Error AbsolutePercent

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

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

1200225

2240220

3300285

4270290

5230250

6260240

7210250

8275240

-

Total

Choosing a MethodForecast Error


Choosing a method forecast error2

MSE = =

MAD = =

MAPE = =

Choosing a MethodForecast Error

Measures of Error

RSFE =


Choosing a method forecast error3

Running SumMean Absolute

of Forecast ErrorsDeviation

Method(RSFE - bias)(MAD)

Simple moving average

Three-week (n = 3)23.117.1

Six-week (n = 6)69.815.5

Weighted moving average

0.70, 0.20, 0.1014.018.4

Exponential smoothing

 = 0.165.614.8

 = 0.241.015.3

Choosing a MethodForecast Error


Choosing a method tracking signals

RSFE

MAD

Tracking signal =

+2.0 —

+1.5 —

+1.0 —

+0.5 —

0 —

- 0.5 —

- 1.0 —

- 1.5 —

Control limit

Tracking signal

Control limit

|||||

0510152025

Observation number

Choosing a MethodTracking Signals


Choosing a method tracking signals1

RSFE

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

|||||

0510152025

Observation number

Choosing a MethodTracking Signals


Chapter 13 forecasting

Choosing a MethodTracking Signals


Chapter 13 forecasting

Problem 13-2: Historical demand for a product is:

MonthJanFebMarAprMayJun

Demand121115121615

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


Chapter 13 forecasting

Problem 13-15: In 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:

Week123456

Forecast8008509509501,000975

Actual9001,0001,0509009001,100

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


Chapter 13 forecasting

Methods for Stationary Series


Time series methods simple moving averages

450 —

430 —

410 —

390 —

370 —

Patient arrivals

Actual patient

arrivals

||||||

051015202530

Time Series MethodsSimple Moving Averages

Week


Time series methods simple moving averages1

Time Series MethodsSimple Moving Averages

450 —

430 —

410 —

390 —

370 —

Patient

WeekArrivals

1400

2380

3411

Patient arrivals

Actual patient

arrivals

||||||

051015202530

Week


Time series methods simple moving averages2

Time Series MethodsSimple Moving Averages

450 —

430 —

410 —

390 —

370 —

Patient

WeekArrivals

1400

2380

3411

Patient arrivals

F4 =

Actual patient

arrivals

||||||

051015202530

Week


Time series methods simple moving averages3

Time Series MethodsSimple Moving Averages

450 —

430 —

410 —

390 —

370 —

Patient

WeekArrivals

2380

3411

4415

Patient arrivals

F5 =

Actual patient

arrivals

||||||

051015202530

Week


Time series methods simple moving averages4

450 —

430 —

410 —

390 —

370 —

3-week MA

forecast

Patient arrivals

Actual patient

arrivals

||||||

051015202530

Week

Time Series MethodsSimple Moving Averages


Time series methods simple moving averages5

450 —

430 —

410 —

390 —

370 —

6-week MA

forecast

3-week MA

forecast

Patient arrivals

Actual patient

arrivals

||||||

051015202530

Time Series MethodsSimple Moving Averages

Week


Chapter 13 forecasting

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

t-10.70

t-20.20

t-30.10

Time Series MethodsWeighted Moving Average

450 —

430 —

410 —

390 —

370 —

3-week MA

forecast

Weighted Moving Average

Patient arrivals

F4 =

Actual patient

arrivals

||||||

051015202530

Week


Time series methods weighted moving average1

Assigned weights

t-10.70

t-20.20

t-30.10

Time Series MethodsWeighted Moving Average

450 —

430 —

410 —

390 —

370 —

3-week MA

forecast

Weighted Moving Average

Patient arrivals

F5 =

Actual patient

arrivals

||||||

051015202530

Week


Time series methods exponential smoothing

Time Series MethodsExponential Smoothing

450 —

430 —

410 —

390 —

370 —

Exponential Smoothing

 = 0.10

Ft = At-1 + (1 - )Ft - 1

Patient arrivals

||||||

051015202530

Week


Time series methods exponential smoothing1

Time Series MethodsExponential 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

||||||

051015202530

Week


Time series methods exponential smoothing2

Time Series MethodsExponential 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 =

||||||

051015202530

Week


Time series methods exponential smoothing3

Time Series MethodsExponential Smoothing

450 —

430 —

410 —

390 —

370 —

Exponential Smoothing

 = 0.10

Ft = At + (1 - )Ft - 1

F4 =

A4 = 415

Patient arrivals

F5 =

||||||

051015202530

Week


Time series methods exponential smoothing4

450 —

430 —

410 —

390 —

370 —

Patient arrivals

||||||

051015202530

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


Chapter 13 forecasting

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!


Chapter 13 forecasting

Problem 13-20: Your 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:

Month1 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?


Trend based methods

Trend-Based Methods


Chapter 13 forecasting

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

Dependent variable

X

Independent variable

Linear Regression


Linear regression1

Regression

equation:

Y = a + bX

Y

Dependent variable

X

Independent variable

Linear Regression


Linear regression2

Regression

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

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,

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

SalesAdvertising

Month(000 units)(000 $)

12642.5

21161.3

31651.4

41011.0

52092.0

Linear Regression


Linear regression6

Sales, yAdvertising, x

Month(000 units)(000 $)

12642.5

21161.3

31651.4

41011.0

52092.0

xy - nxy

x2 - n(x)2

a = y - bx

b =

Linear Regression


Linear regression7

Sales, yAdvertising, x

Month(000 units)(000 $)xyx 2

12642.5

21161.3

31651.4

41011.0

52092.0

Total

y= x =

xy - nxy

x 2 - nx 2

a = y - bx

b =

Linear Regression


Chapter 13 forecasting

300 —

250 —

200 —

150 —

100 —

50

Sales (000s)

||||

1.01.52.02.5

b = 109.229

Y =


Linear regression8

Sales, yAdvertising, x

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

12642.5660.06.25

21161.3150.81.69

31651.4231.01.96

41011.0101.01.00

52092.0418.04.00

Total8558.21560.814.90

y = 171x = 1.64

nxy - x y

[nx 2 -(x) 2][ny 2 - (y) 2]

r =

Linear Regression


Chapter 13 forecasting

Sales, yAdvertising, x

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

12642.5660.06.2569,696

21161.3150.81.6913,456

31651.4231.01.9627,225

41011.0101.01.0010,201

52092.0418.04.0043,681

Total8558.21560.814.90164,259

y= 171x = 1.64

r = 0.98 r 2 = 0.96

Linear Regression


Linear regression9

Linear Regression

Forecast for Month 6:

Advertising expenditure = $1750

Y =


Time series methods linear regression analysis

Time Series MethodsLinear Regression Analysis

80 —

70 —

60 —

50 —

40 —

30 —

Yn = a + bXn

where

Xn = Weekn

Patient arrivals

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

0123456789101112131415

Week


Time series methods linear regression analysis1

Time Series MethodsLinear Regression Analysis

80 —

70 —

60 —

50 —

40 —

30 —

Yn = a + bXn

where

Xn = Weekn

Patient arrivals

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

0123456789101112131415

Week


Chapter 13 forecasting

Time Series MethodsLinear Regression Analysis

  • Standard error of estimate is computed as follows:


Chapter 13 forecasting

Time Series MethodsLinear 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?


Chapter 13 forecasting

Time Series MethodsDouble Exponential Smoothing

  • The method uses two smoothing constants  and 


Chapter 13 forecasting

A Comparison of Methods

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


Time series methods seasonal influences

QuarterYear 1Year 2Year 3Year 4

14570100100

2335370585725

35205908301160

4100170285215

Total1000120018002200

Average250300450550

Time Series MethodsSeasonal Influences


Time series methods seasonal influences1

QuarterYear 1Year 2Year 3Year 4

14570100100

2335370585725

35205908301160

4100170285215

Total1000120018002200

Average250300450550

Actual Demand

Average Demand

Seasonal Index =

Time Series MethodsSeasonal Influences


Time series methods seasonal influences2

QuarterYear 1Year 2Year 3Year 4

14570100100

2335370585725

35205908301160

4100170285215

Total1000120018002200

Average250300450550

Seasonal Index = =

Time Series MethodsSeasonal Influences


Time series methods seasonal influences3

QuarterYear 1Year 2Year 3Year 4

145/250 = 70100100

2335370585725

35205908301160

4100170285215

Total1000120018002200

Average250300450550

Seasonal Index = =

Time Series MethodsSeasonal Influences


Time series methods seasonal influences4

Quarter Year 1 Year 2 Year 3 Year 4

145/250 = 0.1870/300 = 0.23100/450 = 0.22100/550 = 0.18

2335/250 = 1.34370/300 = 1.23585/450 = 1.30725/550 = 1.32

3520/250 = 2.08590/300 = 1.97830/450 = 1.841160/550 = 2.11

4100/250 = 0.40170/300 = 0.57285/450 = 0.63215/550 = 0.39

Time Series MethodsSeasonal Influences


Time series methods seasonal influences5

Quarter Year 1 Year 2 Year 3 Year 4

145/250 = 0.1870/300 = 0.23100/450 = 0.22100/550 = 0.18

2335/250 = 1.34370/300 = 1.23585/450 = 1.30725/550 = 1.32

3520/250 = 2.08590/300 = 1.97830/450 = 1.841160/550 = 2.11

4100/250 = 0.40170/300 = 0.57285/450 = 0.63215/550 = 0.39

QuarterAverage 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

Average Quarterly Demand = 2600/4 = 650

QuarterAverage Seasonal IndexForecast

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

2

3

4

Time Series MethodsSeasonal Influences


Seasonal influences

(a) Multiplicative influence

Demand

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

0245810121416

Period

Seasonal Influences


Seasonal influences1

(b) Additive influence

Demand

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

0245810121416

Period

Seasonal Influences


Chapter 13 forecasting

Time Series MethodsSeasonal 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


Chapter 13 forecasting

Time Series MethodsSeasonal 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


Chapter 13 forecasting

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

YearSeasonActual Demand

2004Spring205

Summer140

Fall375

Winter575

2005Spring475

Summer275

Fall685

Winter965


Chapter 13 forecasting

Reading and Exercises

  • Chapter 13 pp. 518-539

  • Problems 1, 7, 13, 14,16


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