Textbook: pp. 165-202

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Chapter 5: Forecasting

Textbook: pp. 165-202

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Learning Objectives

After completing this chapter, students will be able to:

• Understand and know when to use various families of

forecasting models

• Compare moving averages, exponential smoothing, and

other time-series models

• Calculate measures of forecast accuracy

• Apply forecast models for random variations

• Apply forecast models for trends and random variations

• Manipulate data to account for seasonal variations

• Apply forecast models for trends, seasonal variations, and

random variations

• Explain how to monitor and control forecasts

3

• Main purpose of forecasting

Reduce uncertainty and make better estimates of

what will happen in the future

• Subjective methods

Seat-of-the pants methods, intuition, experience

• More formal quantitative and qualitative techniques

Introduction

4

Forecasting Models

5

• Incorporate judgmental or subjective factors

o Useful when subjective factors are important or

accurate quantitative data is difficult to obtain

• Common qualitative techniques

1. Delphi method

2. Jury of executive opinion

3. Sales force composite

4. Consumer market surveys

Qualitative Models (1 of 3)

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• Delphi Method

o Iterative group process

o Respondents provide input to decision makers

o Repeated until consensus is reached

• Jury of Executive Opinion

o Collects opinions of a small group of high-level

managers

o May use statistical models for analysis


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• Sales Force Composite

o Allows individual salespersons estimates

o Reviewed for reasonableness

o Data is compiled at a district or national level

• Consumer Market Survey

o Information on purchasing plans solicited from

customers or potential customers

o Used in forecasting, product design, new product

planning


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• Predict the future based on the past

• Uses only historical data on one variable

• Extrapolations of past values of a series

• Ignores factors such as

o Economy

o Competition

o Selling price

Time-Series Models

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• Sequence of values recorded at successive intervals of

time

• Four possible components

o Trend (T)

o Seasonal (S)

o Cyclical (C)

o Random (R)

Components of a Time Series (1 of 3)

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Scatter Diagram for Four Time Series of Quarterly Data:


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Scatter Diagram of a Time Series with Cyclical and

Random Components:


12

Two basic forms:

• Multiplicative

Demand = T × S × C × R

• Additive

Demand = T + S + C + R

• Combinations are possible

Time-Series Models

Trend (T); Seasonal (S); Cyclical (C); Random (R)

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• Compare forecasted values with actual values

o See how well one model works

o To compare models

Forecast error = Actual value − Forecast value

• Measure of accuracy

o Mean absolute deviation (MAD):

Measures of Forecast Accuracy (1 of 5)

forecast errorMAD

n

14

Computing the Mean Absolute Deviation (MAD).


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Computing the Mean Absolute Deviation (MAD):


• Forecast based on naïve model

• No attempt to adjust for time series components

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Computing the Mean Absolute Deviation (MAD):


forecast error 160MAD 17.8

9n

17

Other common measures:

o Mean squared error (MSE)

o Mean absolute percent error (MAPE)

o Bias is the average error


2(error)MSE

n

error

actualMAPE 100%

n

18

• No other components are present

• Averaging techniques smooth out forecasts

o Moving averages

o Weighted moving averages

o Exponential smoothing

Forecasting Random Variations

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• Used when demand is relatively steady over time

o The next forecast is the average of the most recent n

data values from the time series

o Smooths out short-term irregularities in the data

series

Moving Averages (1 of 2)

Sum of demands in previous periodsMoving average forecast =

n

n

20

Mathematically:

where Ft+1 = forecast for time period t + 1

Yt = actual value in time period t

n = number of periods to average

Moving Averages (12 of 2)

1 11

...t t t nt

Y Y YF

n

21

• Wallace Garden Supply wants to forecast demand for its

Storage Shed

o Collected data for the past year

o Use a three-month moving average (n = 3)

Wallace Garden Supply (1 of 4)

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Wallace Garden Supply --- Shed Sales (2 of 4)

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• Weighted moving averages use weights to put more

emphasis on previous periods

• Often used when a trend or other pattern is emerging

Mathematically

where wi = weight for the ith observation

Weighted Moving Averages

1

(Weight in period )(Actual value in period )

(Weights)t

i iF

1 2 1 11

1 2

...

...t t n t n

t

n

w Y w Y w YF

w w w

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• Use a 3-month weighted moving average model to

forecast demand

Weighting scheme:


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Weighted Moving Average Forecast for Wallace Garden

Supply:


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Exponential smoothing:

o A type of moving average

o Easy to use

o Requires little record keeping of data

New forecast = Last period’s forecast

+ α(Last period’s actual demand

− Last period’s forecast)

α is a weight (or smoothing constant) with a value 0 ≤ α ≤ 1

Exponential Smoothing (1 of 2)

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Mathematically

where Ft+1 = new forecast (for time period t + 1)

Yt = pervious forecast (for time period t)

α = smoothing constant (0 ≤ α ≤ 1)

Yt = pervious period’s actual demand

Exponential Smoothing (2 of 2)

1 ( )t t t tF F Y F

The idea is simple – the new estimate is the old estimate plus some fraction of the error in the last period!

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• In January, February’s demand for a certain car model

was predicted to be 142

• Actual February demand was 153 autos

• Using a smoothing constant of α = 0.20, what is the

forecast for March?

New forecast (for March demand) = 142 + 0.2(153 − 142)

= 144.2 or 144 autos

If actual March demand = 136

New forecast (for April demand) = 144.2 + 0.2(136 − 144.2)

= 142.6 or 143 autos

Exponential Smoothing Example (1 of 2)

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• Selecting the appropriate value for α is key to obtaining

a good forecast

• The objective is always to generate an accurate forecast

• The general approach is to develop trial forecasts with

different values of α and select the α that results in the

lowest MAD

Exponential Smoothing Example (2 of 2)

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Port of Baltimore Exponential Smoothing:

• Forecasts for α = 0.10 and α = 0.50

Port of Baltimore Example (1 of 2)

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Absolute Deviations and MADs for the Port of Baltimore

Example:

Port of Baltimore Example (2 of 2)

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Selecting the Forecasting Model in Wallace Garden

Supply Problem:

Using Software (1 of 7)

PROGRAMME 5.1A

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Initialising Excel QM Spreadsheet for Wallace Garden

Supply Problem:


PROGRAMME 5.1B

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Excel QM Output for Wallace Garden Supply Problem:


PROGRAMME 5.1C

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Selecting Time-Series Analysis in QM for Windows in the

Forecasting Module:


PROGRAMME 5.2A

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Entering Data for Port of Baltimore Example in QM for

Windows:


PROGRAMME 5.2B

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Selecting the Model and Entering Data for Port of

Baltimore Example in QM for Windows:


PROGRAMME 5.2C

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Output for Port of Baltimore Example in QM for Windows:


PROGRAMME 5.2D

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• Exponential smoothing does not respond to trends

• A more complex model can be used

• The basic approach

o Develop an exponential smoothing forecast

o Adjust it for the trend

Forecasting Models – Trend and Random

Variations

40

• The equation for the trend correction uses a new

smoothing constant β

• Ft and Tt must be given or estimated

• Three steps in developing FITt

Step 1: Compute smoothed forecast Ft+1

Smoothed forecast = Previous forecast including

trend + a(Last error)

Exponential Smoothing with Trend (1 of 2)

1 ( )t t t tF FIT Y FIT

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Step 2: Update the trend (Tt +1) using

Smoothed forecast = Previous forecast including trend +

b(Error or excess in trend)

Step 3: Calculate the trend-adjusted exponential smoothing

forecast (FITt +1) using

Forecast including trend (FITt+1) = Smoothed forecast (Ft+1) +

Smoothed trend (Tt+1)

Exponential Smoothing with Trend (2 of 2)

1 1( )t t t tT T F FIT

1 1 1t t tFIT F T

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• A high value of β makes the forecast more responsive

to changes in trend

• A low value of β gives less weight to the recent trend

and tends to smooth out the trend

• Values are often selected using a trial-and-error

approach based on the value of the MAD for different

values of β

Selecting a Smoothing Constant

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• Demand for electrical generators from 2007 – 2013

o Midwest assumes F1 is perfect, T1 = 0, α = 0.3, β = 0.4

Midwestern Manufacturing’s Demand:

Midwestern Manufacturing (1 of 6)

FIT1 =F1 +T1 = 74+0 = 74

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For 2008 (time period 2)

Step 1: Compute Ft+1

F2 = FIT1 + α(Y1 − FIT1)

= 74 + 0.3(74 − 74) = 74

Step 2: Update the trend

T2 = T1 + β(F2 − FIT1)

= 0 + .4(74 − 74) = 0


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Step 3: Calculate the trend-adjusted exponential

smoothing forecast (Ft+1) using

FIT2 = F2 + T2

= 74 + 0 = 74


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For 2009 (time period 3)

Step 1: F3 = FIT2 + α(Y2 − FIT2)

= 74 + 0.3(79 − 74) = 75.5

Step 2: T3 = T2 + .4(F3 − FIT2)

= 0 + .4(75.5 − 74) = 0.6

Step 3: FIT3 = F3 + T3

= 75.5 + 0.6 = 76.1


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Midwestern Manufacturing Exponential Smoothing with

Trend Forecasts:


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Output from Excel QM in Excel 2016 for Trend-Adjusted

Exponential Smoothing Example:


PROGRAMME 5.3

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• Fits a trend line to a series of historical data points

• Projected into the future for medium- to long-range

forecasts

• Trend equations can be developed based on

exponential or quadratic models

• Linear model developed using regression analysis is

simplest

Trend Projections (1 of 2)

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Mathematical formula:

where Ŷ = predicted value

b0 = intercept

b1 = slope of the line

X = time period (i.e., X = 1, 2, 3, …, n)

Trend Projections (2 of 2)

0 1Y b b X

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• Based on least squares regression, the forecast

equation is

• Year 2014 is coded as X = 8

(sales in 2014)= 56.71 + 10.54(8)

= 141.03, or 141 generators

• For X = 9

(sales in 2015)= 56.71 + 10.54(9)

= 151.57, or 152 generators


ˆ 56.71 10.54XY

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Output from Excel QM in Excel 2016 for Trend Line

Example:


PROGRAMME 5.4

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Output from QM for Windows for Trend Line Example:


PROGRAMME 5.5

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Generator Demand and Projections for Next Three Years

Based on Trend Line:


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• Recurring variations over time may indicate the need for

seasonal adjustments in the trend line

• A seasonal index indicates how a particular season

compares with an average season

o An index of 1 indicates an average season

o An index > 1 indicates the season is higher than average

o An index < 1 indicates a season lower than average

Seasonal Variations

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• Deseasonalised data is created by dividing each

observation by the appropriate seasonal index

• Once deseasonalised forecasts have been developed,

values are multiplied by the seasonal indices

• Computed in two ways

o Overall average

o Centered-moving-average approach

Seasonal Indices

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• Divide average value for each season by the average of

all data

o Telephone answering machines at Eichler Supplies

o Sales data for the past two years for one model

o Create a forecast that includes seasonality

Seasonal Indices with No Trend (1 of 3)

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Answering Machine Sales and Seasonal Indices:


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Calculations for the seasonal indices:


60

• Changes could be due to trend, seasonal, or random

• Centered moving average (CMA) approach prevents

trend being interpreted as seasonal

• Turner Industries sales contain both trend and seasonal

components

Seasonal Indices with Trend (1 of 2)

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Steps in CMA:

1. Compute the CMA for each observation (where

possible)

2. Compute the seasonal ratio = Observation÷CMA for

that observation

3. Average seasonal ratios to get seasonal indices

4. If seasonal indices do not add to the number of

seasons, multiply each index by (Number of

seasons)÷(Sum of indices)

Seasonal Indices with Trend (2 of 2)

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Quarterly Sales ($1,000,000s) for Turner Industries:

Turner Industries (1 of 7)

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Quarterly Sales ($1,000,000s) for Turner Industries:


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• To calculate the CMA for quarter 3 of year 1, compare

the actual sales with an average quarter centred on that

time period

• Use 1.5 quarters before quarter 3 and 1.5 quarters after

quarter 3

o Take quarters 2, 3, and 4 and one half of quarters 1,

year 1 and quarter 1, year 2


CMA(q3,y1)=0.5(108)+125+150+141+0.5(116)

4=132.0

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Compare the actual sales in quarter 3 to the CMA to find

the seasonal ratio:


Seasonal ratio =Sales in quarter 3

CMA=

150

132.0=1.136

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Centered Moving Averages and Seasonal Ratios for

Turner Industries:


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The two seasonal ratios for each quarter are averaged to

get the seasonal index:

Index for quarter 1 = I1 = (0.851 + 0.848)÷2 = 0.85





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Scatterplot of Turner Industries Sales Data and Centered

Moving Average:


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• Decomposition – isolating linear trend and seasonal

factors to develop more accurate forecasts

Five steps to decomposition:

1. Compute seasonal indices using CMAs.

2. Deseasonalise the data by dividing each number by its

seasonal index

3. Find the equation of a trend line using the

deseasonalised data

4. Forecast for future periods using the trend line

5. Multiply the trend line forecast by the appropriate

seasonal index

Trend, Seasonal, and Random Variations

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Deseasonalised Data for Turner Industries (1 of 4)

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• Find a trend line using the deseasonalised data where X

= time

b1 = 2.34 b0 = 124.78

• Develop a forecast for quarter 1, year 4 (X = 13) using

this trend and multiply the forecast by the appropriate

seasonal index

Deseasonalised Data (2 of 4)

Y =124.78+2.34X

Y =124.78+ 2.34(13)

=155.2 (before seasonality adjustment)

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Including the seasonal index:

Y ´ I1 =155.2´0.85 =131.92

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Scatterplot of Turner Industries Original Sales Data and

Deseasonalised Data:


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Turner Industry Forecasts for Four Quarters of Year 4:


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Scatterplot of Turner Industries’ Original Sales Data and

Deseasonalised Data with Unadjusted and Adjusted

Forecasts:


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QM for Windows Input Screen for Turner Industries

Example:


PROGRAMME 5.6A

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QM for Windows Output Screen for Turner Industries

Example:


PROGRAMME 5.6B

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• Multiple regression can be used to forecast both trend

and seasonal components

o One independent variable is time

o Dummy independent variables are used to represent the

seasons

o An additive decomposition model

where X1 = time period

X2 = 1 if quarter 2, 0 otherwise



Using Regression with Trend and Seasonal (1 of 5)

Y = a+b1X1 +b2X2 +b3X3 +b4X4

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Excel QM Multiple Regression Initialisation Screen for

Turner Industries:


PROGRAMME 5.7A

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Excel QM Multiple Regression Output Screen for Turner

Industries:


PROGRAMME 5.7B

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• Regression equation

• Forecasts for first two quarters next year

Y =104.1 + 2.3X1 + 15.7X2 + 38.7X3 + 30.1X4

Y =104.1 + 2.3(13) + 15.7(0) + 38.7(0) + 30.1(0) =134

Y =104.1 + 2.3(14) + 15.7(1) + 38.7(0) + 30.1(0) =152


82

• Different from the results using the multiplicativedecomposition method

• Use MAD or MSE to determine the best model!


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• Tracking signal measures how well a forecast predicts

actual values

o Running sum of forecast errors (RSFE) divided by

the MAD

Monitoring and Controlling Forecasts (1 of 3)

84

• Positive tracking signals indicate demand is greater

than forecast

• Negative tracking signals indicate demand is less than

forecast

• A good forecast will have about as much positive error

as negative error

• Problems are indicated when the signal trips either the

upper or lower predetermined limits

• Choose reasonable values for the limits


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Plot of Tracking Signals:


86

• Quarterly sales of croissants (in thousands)

For Period 6:

Kimball’s Bakery Example

MAD =forecast errorå

n=

85

6=14.2

Tracking signal =RSFE

MAD=

35

14.2= 2.5 MADs

87

• Computer monitoring of tracking signals and self-

adjustment if a limit is tripped

• In exponential smoothing, the values of α and β are

adjusted when the computer detects an excessive

amount of variation

Adaptive Smoothing

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• End of chapter self-test 1-14

(pp. 243-244)

Compile all answers into one

document and submit at the

beginning of the next lecture!

On the top of the document,

write your Pinyin-Name and

Student ID.

• Please read Chapter 6!

Homework --- Chapter 5

Documents

Textbook: pp. 165-202