Chapter 5: Forecasting - UWCENTRE

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Chapter 5: Forecasting

Textbook: pp. 165-202

Every day, managers make

decisions without knowing what will

happen in the future …

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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

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Main purpose of forecasting:

Managers are trying to 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 (e.g. least squares regression

analysis, trend projections …) and qualitative

techniques

Introduction

4

Forecasting Models

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• 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)

6

• 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

Time-Series Models

• If we are forecasting weekly sales

for fidget spinners, we use the past

weekly sales for fidget spinners in

making the forecast for future sales

– ignoring other factors such as

the economy, competition, and

even the selling price of the fidget

spinners.

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• Sequence of values recorded at successive

intervals of time (e.g. weekly sales of Lenovo tablet

computers; quarterly earnings reports of Air China …)

• Four possible components

o Trend (T)

o Seasonal (S)

o Cyclical (C)

o Random (R)

Components of a Time Series (1 of 4)

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1. Trend (T) is the gradual upward or downward

movement of the data over time.

2. Seasonality (S) is a pattern of the demand

fluctuation above or below the trend line that

repeats at regular intervals.

3. Cycles (C) are patterns in annual data that occur

every several years. They are usually tied into the

business cycle (used only in very long forecasts!)

4. Random variations (R) are “blips” in the data

caused by chance and unusual situations; they

follow no discernible pattern.


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Scatter Diagram for Four Time Series of Quarterly Data:


12

Scatter Diagram of a Time Series with Cyclical and

Random Components:


13

Two basic forms:

• Multiplicative (multiplies the components to provide

an estimate)

Demand = T × S × C × R

• Additive (adds the components together to provide

an estimate)

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

• One measure of accuracy is the

o Mean absolute deviation (MAD):

Measures of Forecast Accuracy (1 of 4)

forecast errorMAD

n

deviation

sum of the absolute

values of the

individual forecast

errors

number

of errors (n)

Naïve model – does not attempt to address any of the components of a time series. A

naïve forecast for the next time period is the actual value that was observed in the

current time period.

15

Computing the Mean Absolute Deviation (MAD):


forecast errorMAD

n

Absolute value |X| of a real number X is the non-negative value of X without

regard to its sign.

This means that – on average – each forecast missed the

actual value by 17.8 units!

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Computing the Mean Absolute Deviation (MAD):


• Forecast based on naïve model

• No attempt to adjust for time series components

17

Other common measures:

o Mean squared error (MSE)

o Mean absolute percent error (MAPE)

o Bias is the average error and tells whether the

forecast tends to be too high or too low and by how

much*!


2(error)MSE

n

error

actualMAPE 100%

n

* Bias may be negative or positive Not a good measure of the actual size

of the errors because the negative errors can cancel out the positive errors!

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• If all variations in a time series are due to random

variations [no trend / seasonal / cyclical component is

present] some type of averaging or smoothing model

would be appropriate!

• Averaging techniques smooth out forecasts and will

not be too heavily influenced by random variations:

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

Example “Four-month moving average”: Found by summing the demand

during the past four months and dividing by 4. With each passing month, the

most recent month’s data are added to the sum of the previous three months’

data, and the earliest month is dropped smoothes out irregularities!

Moving Averages (1 of 3)

Sum of demands in previous periodsMoving average forecast =

n

n

n = number of periods to average

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Mathematically:

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

Yt = actual value in time period t



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) and

forecast sales for the next January!

Wallace Garden Supply (1 of 4)

1 11

...t t t nt

Y Y YF

n


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

1 11

...t t t nt

Y Y YF

n


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

1 11

...t t t nt

Y Y YF

n


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• A simple moving average gives the same weight to

each of the past observations being used to develop

the forecast.


1 11

...t t t nt

Y Y YF

n

25

• 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:


1 2 1 11

1 2

...

...t t n t n

t

n

w Y w Y w YF

w w w

Weights of 3 for the most recent observation, 2 for the next observation,

and 1 for the most distant observation. The sum of weights is 6!

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Supply:


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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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)

Ft = 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!

Smoothing constant (α), can be changed to give more weight to recent

data when the value is high or more weight to past data when it is low!

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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) =

Exponential Smoothing Example (1 of 2)

1 ( )t t t tF F Y F

Smoothing constant (α) is low --- more weight to past data is given.

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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) =


1 ( )t t t tF F Y F

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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


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• Selecting the appropriate value for our smoothing

constant α is key to obtaining a good forecast

• The objective is always to generate an accurate

forecast (only the correct α delivers this result!)

• The general approach is to develop trial forecasts

with different values of α and select the α that

results in the lowest Mean Absolute Deviation

(MAD).


forecast errorMAD

n

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Let us apply this concept with a trial-and-error testing of

two values of α = 0.1 and = 0.5 in an example.

The port of Baltimore has unloaded large quantities of

grain from ships during the past eight quarters.

The port’s operations manager wants to test the use of

exponential smoothing to see how well the technique

works in predicting tonnage unloaded.

He assumes that the forecast of grain unloaded in the first

quarter was 175 tons.

exponential smoothing model is used for the first time ---

a previous value for a forecast must be assumed!

Port of Baltimore Example (1 of 3)

1 ( )t t t tF F Y F

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Port of Baltimore Exponential Smoothing:

• Forecasts for α = 0.10 and α = 0.50


1 ( )t t t tF F Y F

forecast errorMAD

n

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Absolute Deviations and MADs for the Port of Baltimore

Example:


n = number of errors (n)

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Selecting the Forecasting Model in Wallace Garden

Supply Problem:

Using Software (1 of 7)

PROGRAMME 5.1A

Weighted moving average

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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Supply:


1 2 1 11

1 2

...

...t t n t n

t

n

w Y w Y w YF

w w w

Weights of 3 for the most recent observation, 2 for the next observation,

and 1 for the most distant observation. The sum of weights is 6!

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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

If there is a trend present in a time series, the

forecasting model must account for this and cannot

simply average past values!

• A more complex model can be used

• The basic idea is

o Develop an exponential smoothing forecast, and

o Adjust it for the trend!

Forecasting Models – Trend and Random

Variations

Trend (T) is the gradual upward or downward movement

of the data over time.

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• 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 + α(Last error)

Exponential Smoothing with Trend (1 of 2)

1 ( )t t t tF FIT Y FIT

forecast including trend

Smoothing constant for forecast: alpha (α)

Smoothing constant for trend: beta (β)

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

Smoothed forecast = Previous forecast including trend +

β(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 Mean Absolute

Deviation (MAD) for different values of β

Selecting a Smoothing Constant




50

• 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)

=

Step 2: Update the trend

T2 = T1 + β(F2 − FIT1)

=


F1 is perfect, T1 = 0, α = 0.3, β = 0.4



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F2 = FIT1 + α(Y1 − FIT1)

= 74 + 0.3(74 − 74) = 74


T2 = T1 + β(F2 − FIT1)

=


F1 is perfect, T1 = 0, α = 0.3, β = 0.4



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F2 = FIT1 + α(Y1 − FIT1)

= 74 + 0.3(74 − 74) = 74


T2 = T1 + β(F2 − FIT1)

= 0 + .4(74 − 74) = 0


F1 is perfect, T1 = 0, α = 0.3, β = 0.4

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Step 3: Calculate the trend-adjusted exponential

smoothing forecast (Ft+1) using

FIT2 = F2 + T2

=


1 1 1t t tFIT F T

55

Step 3: Calculate the trend-adjusted exponential

smoothing forecast (Ft+1) using

FIT2 = F2 + T2

= 74 + 0 = 74


FIT2 = 74, T2 = 0

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Step 1: F3 = FIT2 + α(Y2 − FIT2)

=

Step 2: T3 = T2 + β(F3 − FIT2)

=

Step 3: FIT3 = F3 + T3

=


F1 is perfect, T1 = 0, α = 0.3, β = 0.4

FIT2 = 74, T2 = 0



1 1 1t t tFIT F T

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Step 1: F3 = FIT2 + α(Y2 − FIT2)

= 74 + 0.3(79 − 74) = 75.5

Step 2: T3 = T2 + β(F3 − FIT2)

= 0 + .4(75.5 − 74) = 0.6

Step 3: FIT3 = F3 + T3

= 75.5 + 0.6 = 76.1


F1 is perfect, T1 = 0, α = 0.3, β = 0.4

FIT2 = 74, T2 = 0



1 1 1t t tFIT F T

58

Midwestern Manufacturing Exponential Smoothing with

Trend Forecasts:


59

Output from Excel QM in Excel 2016 for Trend-Adjusted

Exponential Smoothing Example:


PROGRAMME 5.3

60

• Fits a trend line to a series of

historical data points – and

then

• projects the line 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)

0 1Y b b X

A trend line is simply a linear regression

equation in which the independent variable

(X) is the time period.

61

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

64

Output from QM for Windows for Trend Line Example:


PROGRAMME 5.5

65

Generator Demand and Projections for Next Three Years

Based on Trend Line:


66

Example:

• Demand for coal and fuel oil usually peaks during the

cold winter months.

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

Analysing data in monthly or quarterly terms makes it

easy to spot seasonal patterns!

When no trend is present, the index

can be found by dividing the average

value for a particular season by the

average of all the data

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• Each observation in the time series is divided by the

appropriate seasonal index to remove the impact of the

seasonality Deseasonalised data is created!

• Once deseasonalised forecasts have been developed,

values are multiplied by the seasonal indices

Seasonal indices are computed in two ways:

o Overall average (when no trend is present)

o Centered-moving-average approach (when trend is

present)

Seasonal Indices

68

• 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)

An index of 1 means the season is average!

69

Answering Machine Sales and Seasonal Indices:


First: The average demand

in each month is computed.

(80 + 100) /2 = = 90 / 94

70

Calculations for the seasonal indices:


An index of 1 means the season is average!

Let‘s now suppose we expected the third

year‘s annual demand to be 1,200 units!

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• When both trend and seasonal components are present

in a time series, a change from one month to the next

could be due to a trend, to a seasonal variation, or

simply to random fluctuations.

• Centered moving average (CMA) approach prevents

trend being interpreted as seasonal

New example:

• Turner Industries sales contain both trend and

seasonal components

Seasonal Indices with Trend (1 of 2)

72

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

Can you identify any trend or seasonal components?

Turner Industries (1 of 7)

73

Steps in Centered moving average (CMA):

1. Compute the CMA for each observation (where

possible) … e.g. quarter 3 of year 1

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)

74

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


Contains trend and seasonal components!

75

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

sales (150) with an average quarter centred on that time period

Centre is Q3

• We have a total of 4 quarters (1 year data) – but we need an

equal number of quarters before and after the quarter 3 so the

trend is averaged out!

• 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

76

Compare the actual sales in quarter 3 to the centered

moving average (CMA) to find the seasonal ratio:

CMA (q3, y1) = 132


Seasonal ratio =Sales in quarter 3

CMA=

150

132.0=1.136

77

Centered Moving Averages and Seasonal Ratios for

Turner Industries:

Turner Industries (5 of 7) What does this ratio say? Interpretation?

78

Centered Moving Averages and Seasonal Ratios for

Turner Industries:


We now have two seasonal

indices for Q3 (year 1 and year 2)

Sales in Q3 of Year 1 are about 13.6%

higher than an average quarter!

79

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




The sum of these indices should be the number of

seasons (4) since an average season should have

an index of 1.


If the sum were not 4, an adjustment would be made. We would

multiply each index by 4 and divide this by the sum of the indices.

80

Scatterplot of Turner Industries Sales Data and Centered

Moving Average:

Turner Industries (7 of 7) Contrast CMA plot and original data?

Do you see a definite trend?

81

The Decomposition method isolates the 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

82

Deseasonalised Data for Turner Industries (1 of 4)

Steps 1 and 2:

= 108 / 0.85

Compute seasonal indices using CMAs

Deseasonalise the data by dividing each number by its seasonal index

83

• 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)

We use computer

software for this!

84


Including the seasonal index:

85

Scatterplot of Turner Industries Original Sales Data and

Deseasonalised Data:


86

Turner Industry Forecasts for Four Quarters of Year 4:


Y ´ I1 =155.2´0.85 =131.922

87

Scatterplot of Turner Industries’ Original Sales Data and

Deseasonalised Data with Unadjusted and Adjusted

Forecasts:


90

• 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

95

• 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)

After the forecast has been completetd … it is important that it is

not forgotten!

number

of errors (n)

96

• 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 these limits!


97

Plot of Tracking Signals:


Manager defines

acceptable deviation!

98

• Quarterly sales of croissants (in thousands)

For Period 6:

Kimball’s Bakery Example

RSFE = Running sum of forecast errors

MAD = Mean absolute deviation

99


For Period 6:




100


For Period 6:




101


For Period 6:




Quite a good value – we should be well within our forecast!

102

• 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

103

• 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

Chapter 5: Forecasting - UWCENTRE

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