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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 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


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


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

  • Components of demand

  • Evaluation of forecasts

  • Time series: stationary series

  • Time series: trend

    • Linear regression

    • Double exponential smoothing

  • Time series: seasonality


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


Components of Demand

Qantity

Time

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


Components of Demand

Quantity

Time

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


Components of Demand

Year 1

Quantity

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

JFMAMJJASOND

Months

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


Components of Demand

Year 1

Quantity

Year 2

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

JFMAMJJASOND

Months

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


Components of Demand


Components of Demand

Quantity

||||||

123456

Years

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


Components of Demand


Components of Demand

Trend

Demand

Random

movement

Time

Demand

Trend with

seasonal pattern

Time


Snow Skiing

Seasonal

Long term growth trend

Demand for skiing products increased sharply after the Nagano Olympics


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


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


MSE = =

MAD = =

MAPE = =

Choosing a MethodForecast Error

Measures of Error

RSFE =


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


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


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


Choosing a MethodTracking Signals


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


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.


Methods for Stationary Series


450 —

430 —

410 —

390 —

370 —

Patient arrivals

Actual patient

arrivals

||||||

051015202530

Time Series MethodsSimple Moving Averages

Week


Time Series MethodsSimple Moving Averages

450 —

430 —

410 —

390 —

370 —

Patient

WeekArrivals

1400

2380

3411

Patient arrivals

Actual patient

arrivals

||||||

051015202530

Week


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 MethodsSimple Moving Averages

450 —

430 —

410 —

390 —

370 —

Patient

WeekArrivals

2380

3411

4415

Patient arrivals

F5 =

Actual patient

arrivals

||||||

051015202530

Week


450 —

430 —

410 —

390 —

370 —

3-week MA

forecast

Patient arrivals

Actual patient

arrivals

||||||

051015202530

Week

Time Series MethodsSimple Moving Averages


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


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.


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


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 MethodsExponential Smoothing

450 —

430 —

410 —

390 —

370 —

Exponential Smoothing

 = 0.10

Ft = At-1 + (1 - )Ft - 1

Patient arrivals

||||||

051015202530

Week


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 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 MethodsExponential Smoothing

450 —

430 —

410 —

390 —

370 —

Exponential Smoothing

 = 0.10

Ft = At + (1 - )Ft - 1

F4 =

A4 = 415

Patient arrivals

F5 =

||||||

051015202530

Week


450 —

430 —

410 —

390 —

370 —

Patient arrivals

||||||

051015202530

Week

Time Series MethodsExponential Smoothing


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: 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


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.


Y

Dependent variable

X

Independent variable

Linear Regression


Regression

equation:

Y = a + bX

Y

Dependent variable

X

Independent variable

Linear Regression


Regression

equation:

Y = a + bX

Y

Actual

value

of Y

Dependent variable

Value of X used

to estimate Y

X

Independent variable

Linear Regression


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


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


SalesAdvertising

Month(000 units)(000 $)

12642.5

21161.3

31651.4

41011.0

52092.0

Linear Regression


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


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


300 —

250 —

200 —

150 —

100 —

50

Sales (000s)

||||

1.01.52.02.5

b = 109.229

Y =


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


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 Regression

Forecast for Month 6:

Advertising expenditure = $1750

Y =


Time Series MethodsLinear Regression Analysis

80 —

70 —

60 —

50 —

40 —

30 —

Yn = a + bXn

where

Xn = Weekn

Patient arrivals

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

0123456789101112131415

Week


Time Series MethodsLinear Regression Analysis

80 —

70 —

60 —

50 —

40 —

30 —

Yn = a + bXn

where

Xn = Weekn

Patient arrivals

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

0123456789101112131415

Week


Time Series MethodsLinear Regression Analysis

  • Standard error of estimate is computed as follows:


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?


Time Series MethodsDouble Exponential Smoothing

  • The method uses two smoothing constants  and 


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


QuarterYear 1Year 2Year 3Year 4

14570100100

2335370585725

35205908301160

4100170285215

Total1000120018002200

Average250300450550

Time Series MethodsSeasonal Influences


QuarterYear 1Year 2Year 3Year 4

14570100100

2335370585725

35205908301160

4100170285215

Total1000120018002200

Average250300450550

Actual Demand

Average Demand

Seasonal Index =

Time Series MethodsSeasonal Influences


QuarterYear 1Year 2Year 3Year 4

14570100100

2335370585725

35205908301160

4100170285215

Total1000120018002200

Average250300450550

Seasonal Index = =

Time Series MethodsSeasonal Influences


QuarterYear 1Year 2Year 3Year 4

145/250 = 70100100

2335370585725

35205908301160

4100170285215

Total1000120018002200

Average250300450550

Seasonal Index = =

Time Series MethodsSeasonal Influences


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


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


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


(a) Multiplicative influence

Demand

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

0245810121416

Period

Seasonal Influences


(b) Additive influence

Demand

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

0245810121416

Period

Seasonal Influences


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


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


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


Reading and Exercises

  • Chapter 13 pp. 518-539

  • Problems 1, 7, 13, 14,16


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