lecture 11 forecasting methods qm

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FORECASTING METHODS

Lecture 11

Lecturer : Dr. Dwayne Devonish

MGMT 2012: Introduction to Quantitative MethodsLearning Objectives

• Students should be able to:

� Define the concept of a forecast and

forecast error

�Calculate different measures of forecast

error

� Apply, evaluate and compare different

methods of forecasting in QM based on

time series data.

Introduction to Forecasting

• A forecast is a statement about the futurevalue of a particular variable (e.g. demand,sales, etc).

• Forecasts provide managers with essentialinformation about the future, even in themidst of uncertainty.

• Managers must rely on forecasts in QM tohelp them make more informed decisionsabout meeting future demand.

Forecasting Benefits• Forecasting helps managers:

�Know what capacity will be needed to makestaffing and equipment decisions,

�Prepare sound budgets for future decisions,

�Determine optimal production and inventorylevels

�Anticipate, plan, and respond to future changesor events

• However, managers or quantitative analystsmust be able to select accurate forecastingmethods in order to enjoy desired benefits ofthese tools.

Forecast Accuracy• Selecting a forecasting method depends on the

accuracy of the method compared to other methods.

• Forecasters need to choose methods which generatethe least amount of error - this represents themethods which are most accurate (betterrepresentation of actual data)

• Forecast error is simply the difference between actualdata values or demand and forecast values.

• Forecast Error = Actual (A) - Forecast (F).

• If an actual demand value was 19 and acorresponding forecast value in the same periodwas 23, the forecast error is - 4.

• Forecast error can be converted to an absoluteerror which is simply the removal of the negativesign from the value thus creating an absolute(non-negative number) which is 4.

• This is called absolute error or [error].

Measures of Forecast Accuracy

Measures of forecast accuracy include:

� Mean Absolute Deviation (MAD)

Mean Squared Error (MSE)

= ∑∑∑∑ |absolute forecast errors|

n

= ∑∑∑∑ (errors)

n

2

MAD = add absolute errors for periods and divide by

number of available periods used in the forecast.

MSE = square each error value for each period, then

sum all squared error values and divide by no. of

available periods.

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CALCULATING MAD AND MSE IN TIME

SERIES EXAMPLE

Actual Forecast |error| error2

JAN 243 250 7 49

FEB 315 320 5 25

MAR 286 275 11 121

APR 256 260 4 16

MAY 241 250 9 81

JUN 298 275 23 529

(9.83)

(136.8)

MAD =

MSE =

Forecasting Methods

• There are three categories of forecasting

methods:

�Time series methods: Highlight patterns in

historical data and then extrapolate this

pattern into the future.

�Causal methods: Focus on the relationships

between variables to forecast future values.

�Qualitative methods: Focus on subjective

judgements and expert opinions to develop

relevant forecasts.

Time Series Methods

• A time series is a time-ordered sequence ofobservations taken at regular intervals(Stevenson, 2009, p.78)

• Time series can be expressed daily, monthly,yearly, etc.

• Specific time series forecasting methods include:

� Naïve method

� Moving averages method

� Exponential method

� Linear trend projection method

Naïve Method

• This method uses a single previous

value of a time series as the basis of

the forecast.

• It is useful for stable time series.

• Most simple and straightforward

forecasting method available.

Yearly Time Series:Naïve Method

Years

Actual (A) Demand

(No. of customer

orders)

Forecast (F)

2000 5 -

2001 9 5

2002 6 9

2003 12 6

2004 7 12

2005 10 7

2006 8 10

Forecast Accuracy for Naïve Method

Yrs A. F. [Error] [Error]2

2000 5 - - -

2001 9 5 4 16

2002 6 9 3 9

2003 12 6 6 36

2004 7 12 5 25

2005 10 7 3 9

2006 8 10 2 4

MAD 3.83 -

MSE 16.5

[ERROR] is the absolute error which is the difference

between actual and forecast. Minus signs are removed.

MAD

SUM

AND

AVERAGE

MSE

SUM

AND

AVERAGE

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Moving Averages Method• This method consists of computing an average of

the most recent n data values in a time series

and using this average for the forecast of the next

period.

• Moving averages methods use more historical

data to forecast future values. These methods are

referred to ‘smoothing methods’.

• Good for stable time series and short-range

forecasts.

• You can use different numbers of past periods

(years, months, etc) in the forecasts.

• Consider a 3 year moving average on these

time series data:

Yearly Time Series: 3yr MA

Years

Actual (A) Demand

(No. of customer

orders)

Forecast (F)

3 Year Moving Average

2000 5 -

2001 9 -

2002 6 -

2003 12 (5 + 9 + 6)/3 = 6.67

2004 7 (9 +6 +12)/3 = 9

2005 10 (6 +12+7)/3 = 8.33

2006 8 (12+7+10)/3 =9.67

Forecast Accuracy for 3 yr MA


2000 5 - - -

2001 9 - - -

2002 6 - - -

2003 12 6.67 5.33 28.44

2004 7 9 2 4

2005 10 8.33 1.67 2.78

2006 8 9.67 1.67 2.78

MAD 2.67

MSE 9.50


between actual and forecast: Minus signs removed.

LET’S TRY A 2 YEAR MOVING

AVERAGE FORECAST

NB:Whereas a 3 year moving average begins on the

4th year given that it uses data from the 3 most

recent years, the 2 year moving average forecast

begins on the 3rd year (it uses the 2 most recent

years)

Forecast Accuracy for 2 yr MA


2000 5 - - -

2001 9 - - -

2002 6 7 1 1

2003 12 7.5 4.5 20.25

2004 7 9 2 4

2005 10 9.5 0.5 .25

2006 8 8.5 0.5 .25

MAD 1.70

MSE 5.15



Exponential Smoothing• Another smoothing method which represents a

weighted average method.

• It uses a weighted average of past data values as

a forecast.

• Each new forecast is based on the previous

forecast plus a percentage of the difference

between the forecast and the actual value in that

previous period (i.e. percentage of forecast error –

not absolute error).

• Next Forecast = Previous forecast + ά (Previous

Actual – Previous Forecast), where:

• (Actual – Forecast) = Error in last period, and ά is

percentage of error (i.e. smoothing constant).

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Smoothing Constant in Exponential

Smoothing• The smoothing constant provides the weight for the

next forecast. It is seen as the percentage of error

from last period forecast which you are planning to

incorporate into a future forecast.

• It ranges from 0 to 1. Higher values make greater

adjustments to the next forecast based on past

forecast errors. That is, higher values suggest that

the forecast will be more sensitive to recent changes

to actual demand values.

• If the constant was zero then,

• Next Forecast = Previous forecast + 0 (A - F)

• We consider no percentage of error from last forecast

period, so the next forecast equals previous forecast.

Exponential Smoothing• Commonly used smoothing constants range

from .05 to .50.

• Lower values work well when the underlyingaverage (or series) tends to be stable, whereashigher values work well when the underlyingaverage is susceptible to change (i.e. volatile).

• Let’s use exponential smoothing forecast, with asmoothing constant of .30 (i.e. 30% of priorforecast error is used for future forecasts).

• NB: For the 1st period, no forecast value isgenerated. However, the forecast value in the2nd period in the time series always reflects theactual demand value in the past period. So ifperiod 1 has an actual demand of 5, then theforecast for period 2 is 5. See next slide:

Exponential Smoothing at 0.3

Years

Actual (A)

Demand

(No. of

customer

orders)

Forecast (F)

(0.3 = ά)

Formula: Next forecast =

Previous forecast + ά (A – F)

2000 5 -

2001 9 5

2002 6 6.2 = 5 + .3 (9 - 5)

2003 12 6.14 = 6.2 + .3 (6 - 6.2)

2004 7 7.90 = 6.14 +.3 (12 - 6.14)

2005 10 7.63 = 7.90 +.3 (7 - 7.90)

2006 8 8.34 = 7.63 +.3 (10 – 7.63)

Forecast Accuracy for Exp. Smoot. (0.3)


2000 5 - - -

2001 9 5 4 16

2002 6 6.2 .2 .04

2003 12 6.14 5.86 34.34

2004 7 7.9 0.9 .81

2005 10 7.63 2.37 5.62

2006 8 8.34 .34 .12

MAD 2.28

MSE 9.49



Linear Trend Forecast Method• This involves the use of linear equation to describe

or assess a trend in a time series.

• A trend is a long-term upward or downwardmovement in a time series.

• The linear trend equation has the form:

• Forecast (ŷ) = a + bt, where a = intercept or thevalue of forecast (ŷ) when t is 0; b is the slope ofthe forecast line, and t = number of periods from t= 0 (i.e. 1999).

• Actual values/orders = y,

• forecasted values/orders = ŷ.

• One can must calculate the ‘b’ and then ‘a’ to beable to compute a forecast for a particular period (t=1,2,3,etc)

Formula for ‘b’ (slope)

Formula for ‘a’ (intercept)

for Linear Trend Equation

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LINEAR TREND PROJECTION

Using the table data, compute t, ty, and

t2

Yrs Actual

y t ty t2

2000 5 1 5 1

2001 9 2 18 4

2002 6 3 18 9

2003 12 4 48 16

2004 7 5 35 25

2005 10 6 60 36

2006 8 7 56 49


Using the table data, compute t, ty, and t2

Yrs Actual

y t ty t2

2000 5 1 5 1

2001 9 2 18 4

2002 6 3 18 9

2003 12 4 48 16

2004 7 5 35 25

2005 10 6 60 36

2006 8 7 56 49

Σ 57 28 240 140

ΣY Σt Σty Σt2







7 x 240 28 57x

7 x 140 282

57

7

.43*

28

7




1680 1596

980 784

= 0.43

57

7

.43

28

7

.43* = 6.42

Linear Trend Equation

Forecast: ŷ = a + bt

a (intercept) = 6.42

b (slope) = .43

So Linear trend equation =

ŷ = 6.42 + .43t

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Calculations of Forecasts

Yrs Actual

y T

Forecast

Y = a + bt

2000 5 1 6.85 = 6.42 + .43(1)

2001 9 2 7.28 = 6.42 +.43 (2)

2002 6 3 7.71 = 6.42 + .43 (3)

2003 12 4 8.14 = 6.42 + .43 (4)

2004 7 5 8.57 = 6.42 + .43 (5)

2005 10 6 9.00 = 6.42 + .43 (6)

2006 8 7 9.43 = 6.42 + .43 (7)

Forecast Accuracy for Linear Trend


2000 5 6.85 1.85 3.42

2001 9 7.28 1.72 2.96

2002 6 7.71 1.71 2.92

2003 12 8.14 3.86 14.90

2004 7 8.57 1.57 2.46

2005 10 9.00 1 1

2006 8 9.43 1.43 2.04

MAD 1.87

MSE 3.52[ERROR] is the absolute error which is the difference


COMPARE FORECASTING METHODS

Forecast

Method

MAD MSE

Naive 3.83 16.5

2 yr MA 1.70 5.15 Lowest

MAD

3 yr MA 2.67 9.50

Exp. Smoothing (0.3) 2.28 9.49

Linear Trend

Projection 1.87 3.52

Lowest

MSE

Final Decision

• The measure of forecast accuracy choseninfluences the forecast method selection.

• One would use the ‘2 year MA’ as theforecasting method of choice if MAD is thepreferred criterion, or one may use linear trendmethod if the MSE is the preferred criterion.

• Let’s say we used the MAD as the criterion, youcan use the 2 year forecast method to predict forthe next year (in this case, year 2007). Tocalculate this forecast for 2007, average the datavalues for 2005 and 2006.

• If we relied on the MSE as the criterion, you willuse the linear trend equation to predict 2007: ŷ =6.42 + .43(8).

2 yr M.A Forecast (based on MAD)

Yrs Actual Forecast

2000 5 -

2001 9 -

2002 6 7

2003 12 7.5

2004 7 9

2005 10 9.5

2006 8 8.5

2007 ?? (10 + 8)/2 = 9

Hence, forecast for 2007 is 9000 orders

based on 2 yr MA.

Linear Trend Forecast (based on MSE)

Yrs Actual

y T

Forecast

Y = a + bt

2000 5 1 6.85 = 6.42 + .43(1)

2001 9 2 7.28 = 6.42 +.43 (2)

2002 6 3 7.71 = 6.42 + .43 (3)

2003 12 4 8.14 = 6.42 + .43 (4)

2004 7 5 8.57 = 6.42 + .43 (5)

2005 10 6 9.00 = 6.42 + .43 (6)

2006 8 7 9.43 = 6.42 + .43 (7)

2007 ? 8 9.86 = 6.42 + .43 (8)

Hence, forecast for 2007 is 9860 orders based on linear

trend forecast.

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Qualitative Forecasting� Delphi Method

interactive group process consisting of obtaining information from a group of respondents through

questionnaires and surveys

� Jury of Executive Opinionobtains opinions of a small group of high-level

managers in combination with statistical models

� Sales Force Compositeallows each sales person to estimate the sales for

his/her region and then compiles the data at a district or national level

� Consumer Market Surveysolicits input from customers or potential

customers regarding their future purchasing plans

lecture 11 forecasting methods qm

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