Forecasting
Chapter 3
McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
ForecastForecast – a statement about the future
value of a variable of interestWe make forecasts about such things as
weather, demand, and resource availabilityForecasts are an important element in making
informed decisions
Instructor Slides 3-2
Two Important Aspects of ForecastsExpected level of demand
The level of demand may be a function of some structural variation such as trend or seasonal variation
AccuracyRelated to the potential size of forecast error
Instructor Slides 3-3
Elements of a Good ForecastThe forecast should be timely should be accurate should be reliable should be expressed in meaningful units should be in writing technique should be simple to understand
and use should be cost effective
Steps in the Forecasting Process1. Determine the purpose of the forecast2. Establish a time horizon3. Select a forecasting technique4. Obtain, clean, and analyze appropriate
data5. Make the forecast6. Monitor the forecast
Features Common to All Forecasts1. Techniques assume some underlying causal
system that existed in the past will persist into the future
2. Forecasts are not perfect3. Forecasts for groups of items are more
accurate than those for individual items4. Forecast accuracy decreases as the forecasting
horizon increases
Instructor Slides 3-6
Forecast Accuracy and Control
Forecast errors should be monitored Error = Actual – Forecast If errors fall beyond acceptable bounds,
corrective action may be necessary
Forecast Accuracy Metrics
n
tt ForecastActualMAD
2
tt
1
ForecastActualMSE
n
100Actual
ForecastActualMAPE
t
tt
n
MAD weights all errors evenly
MSE weights errors according to their squared values
MAPE weights errors according to relative error
Forecast Error CalculationPeriod
Actual(A)
Forecast(F)
(A-F) Error |Error| Error2
1 107 110 -3 3 9
2 125 121 4 4 16
3 115 112 3 3 9
4 118 120 -2 2 4
5 108 109 -1 1 1
AVG(A) 114.6 Sum 13 39
n = 5 n-1 = 4
MAD MSE MAPE=MAD / AVG(A)
= 2.6 = 9.75 =2.6/114.6= 2.27%
Forecasting Approaches Qualitative Forecasting
Qualitative techniques permit the inclusion of soft information such as:Human factorsPersonal opinionsHunches
These factors are difficult, or impossible, to quantify Quantitative Forecasting
Quantitative techniques involve either the projection of historical data or the development of associative methods that attempt to use causal variables to make a forecast
These techniques rely on hard data
Judgmental ForecastsForecasts that use subjective inputs such
as opinions from consumer surveys, sales staff, managers, executives, and expertsExecutive opinionsSales force opinionsConsumer surveysDelphi method
Time-Series ForecastsForecasts that project patterns identified
in recent time-series observationsTime-series - a time-ordered sequence of
observations taken at regular time intervalsAssume that future values of the time-
series can be estimated from past values of the time-series
Time-Series BehaviorsTrendSeasonalityCyclesIrregular variationsRandom variation
Historical Monthly Product Demand Consisting of a Growth Trend, Cyclical Factor, and Seasonal Demand
Exhibit 9.4
Common Types of Trends
Exhibit 9.5a
Common Types of Trends (cont’d)
Exhibit 9.5b
Trends and SeasonalityTrend
A long-term upward or downward movement in dataPopulation shiftsChanging income
Seasonality Short-term, fairly regular variations related to the
calendar or time of day Restaurants, service call centers, and theaters all
experience seasonal demand
3-18
Trend, Cyclical, with Variations
Cycles and VariationsCycle
Wavelike variations lasting more than one yearThese are often related to a variety of economic, political,
or even agricultural conditions
Random Variation Residual variation that remains after all other behaviors
have been accounted for
Irregular variation Due to unusual circumstances that do not reflect typical
behaviorLabor strikeWeather event
Time-Series Forecasting - Naïve ForecastNaïve Forecast
Uses a single previous value of a time series as the basis for a forecastThe forecast for a time period is equal to the
previous time period’s valueCan be used when
The time series is stableThere is a trendThere is seasonality
Time-Series Forecasting - AveragingThese Techniques work best when a
series tends to vary about an averageAveraging techniques smooth variations in the
dataThey can handle step changes or gradual
changes in the level of a seriesTechniques
Moving averageWeighted moving averageExponential smoothing
Moving AverageTechnique that averages a number of the
most recent actual values in generating a forecast
average moving in the periods ofNumber
1 periodin valueActual
average moving period MA
period for timeForecast
where
MA
1
1t
n
tA
n
tF
n
AF
t
t
t
n
iit
t
Forecast Demand Based on a Three- andFive-Week Simple Moving Average
Week Demand Forecast Forecast
(3-week) (5-week)
1 800
2 1400
3 1000
4 1500 (1000+1400+800)/3 =1067
5 1500 (1500+1000+1400)/3 = 1300
6 1300 (1500+1500+1000)/3 = 1333 (1500+1500+1000+1400+ 800)/5 =1240
7 1800 (1300+1500+1500)/3 = 1433 (1300+1500+1500+1000+1400)/5 =1340
8 1700 (1800+1300+1500)/3 = 1533 (1800+1300+1500+1500+1000)/5 =1420
9 1300 1600 (1700+1800+1300+1500+1500)/5 =1560
10 1700 1600 (1300+1700+1800+1300+1500)/5 =1520
11 1700 1567 (1700+1300+1700+1800+1300)/5 =1560
Moving AverageAs new data become available, the
forecast is updated by adding the newest value and dropping the oldest and then recomputing the the average
The number of data points included in the average determines the model’s sensitivityFewer data points used-- more responsiveMore data points used-- less responsive
Forecast Demand Based on a Three- andNine-Week Simple Moving Average
Exhibit 9.6
Moving Average Forecast of Three- andNine-Week Periods versus Actual Demand
Exhibit 9.7
Weighted Moving AverageThe most recent values in a time series
are given more weight in computing a forecastThe choice of weights, w, is somewhat
arbitrary and involves some trial and error
Ft wn At n wn 1At (n 1) ... w1At 1
where
wt weight for period t, wt 1 weight for period t 1, etc.
At the actual value for period t, At 1 the actual value for period t 1, etc.
Exponential SmoothingA weighted averaging method that is
based on the previous forecast plus a percentage of the forecast error
1 1 1
1
1
( )
where
Forecast for period
Forecast for the previous period
=Smoothing constant
Actual demand or sales from the previous period
t t t t
t
t
t
F F A F
F t
F
A
Exponential Smoothing
Saturday Hotel Occupancy ( =0.5) Forecast Period Occupancy Forecast Error t At Ft |At - Ft| 1 79 --- 2 84 79.00 5 3 83 79+.5(84-79)=81.50 or 82 1 4 81 81.5+.5(83-81.5)=82.25 or 82 1 5 98 82.25+.5(81-82.25)=81.63 or 82 16 6 100 81.63+.5(98-81.63)= 89.81 or 90 10 MAD =33/5= 6.6 Forecast Error (Mean Absolute Deviation) = ΣlAt – Ftl / n
The first actual value as the forecast for period 2
17-29
Linear TrendA simple data plot can reveal the
existence and nature of a trendLinear trend equation
Ft a bt
where
Ft Forecast for period t
a Value of Ft at t 0
b Slope of the line
t Specified number of time periods from t 0
Estimating slope and interceptSlope and intercept can be estimated
from historical data
22
or
where
Number of periods
Value of the time series
n ty t yb
n t t
y b ta y bt
n
n
y
Figure 3-9
3-32
Linear Trend Example
Week (t) Sales (y) t2 ty
1 150 1 150
2 157 4 314
3 162 9 486
4 166 16 664
5 177 25 885
t= 15 y= 812 t2=55 (ty)=2499
Linear Trend Example
22
5(2499) 15(812)
5(55) 225
12495 121806.3
275 225
812-6.3(15) = 143.5
5143.5 6.3
n ty t yb
n t t
y b ta
ny t
Linear Trend Example
Substituting values of t into this equation, the forecast for next 2 periods are:
F6= 143.5+6.3 (6) = 181.3
F7= 143.5+6.3 (7) = 187.6
Techniques for SeasonalitySeasonality – regularly repeating movements in
series values that can be tied to recurring eventsExpressed in terms of the amount that actual
values deviate from the average value of a seriesModels of seasonality
AdditiveSeasonality is expressed as a quantity that gets added to
or subtracted from the time-series average in order to incorporate seasonality
MultiplicativeSeasonality is expressed as a percentage of the average
(or trend) amount which is then used to multiply the value of a series in order to incorporate seasonality
Instructor Slides 3-36
Models of Seasonality
Instructor Slides 3-37
Computing Seasonal Relatives Using Simple Average (SA) MethodExample 8A, page 150
Manager of a Call center recorded the volume of calls received between 9 and 10 a.m. for 21 days and wants to obtain a seasonal index for each day for that hour.
Volume Season Overall
Day Week 1 Week 2 Week 3 Average ÷ Average = SA Index
Tues 67 60 64 63.667 ÷ 71.571 = 0.8896
Wed 75 73 76 74.667 ÷ 71.571 = 1.0432
Thurs 82 85 87 84.667 ÷ 71.571 = 1.1830
Fri 98 99 96 97.667 ÷ 71.571 = 1.3646
Sat 90 86 88 88.000 ÷ 71.571 = 1.2295
Sun 36 40 44 40.000 ÷ 71.571 = 0.5589
Mon 55 52 50 52.333 ÷ 71.571 = 0.7312
Overall Avg 71.571 7.0000
Seasonal RelativesSeasonal relatives
The seasonal percentage used in the multiplicative seasonally adjusted forecasting model
Using seasonal relatives To deseasonalize data
Done in order to get a clearer picture of the nonseasonal components of the data series
Divide each data point by its seasonal relative To incorporate seasonality in a forecast
Obtain trend estimates for desired periods using a trend equation
Add seasonality by multiplying these trend estimates by the corresponding seasonal relative
Seasonal Relatives ExampleExample 7, page 149
A coffee shop owner wants to predict quarterly
demand for hot chocolate for periods 9 and 10, which
happen to be the 1st and 2nd quarters of a particular
year. The sales data consist of both trend and
seasonality. The trend portion of demand is projected
using the equation Ft = 124 + 7.5 t. Quarter relatives
are
Q1 = 1.20, Q2 = 1.10, Q3 = 0.75, Q4 = 0.95,
Seasonal Relatives Example (Con’d)Example 7, page 149
Use this information to deseasonalize sales for Q1 through Q8.
Period Quarter Sales ÷ Quarter Relative
= Deseasonalized sales
1 1 158.4 ÷ 1.20 = 132.0
2 2 153.0 ÷ 1.10 = 139.1
3 3 110.0 ÷ 0.75 = 146.7
4 4 146.3 ÷ 0.95 = 154.0
5 1 192.0 ÷ 1.20 = 160.0
6 2 187.0 ÷ 1.10 = 170.0
7 3 132.0 ÷ 0.75 = 176.0
8 4 173.8 ÷ 0.95 = 182.9
Seasonal Relatives Example (Con’d)Example 7, page 149
Use this information to predict for periods 9 and 10.
F9 = 124 +7.5( 9) = 191.5 F10= 124 +7.5(10) = 199.0
Multiplying the trend value by the appropriate quarter relative yields a forecast that includes both trend and seasonality.
Given that t =9 is a 1st quarter and t = 10 is a 2nd quarter.
The forecast demand for period 9 = 191.5(1.20) = 229.8 The forecast demand for period 10 = 199.0(1.10) = 218.9