College of Business Administration
Cal State San Marcos
Production & Operations Management
HTM 305
Dr. M. Oskoorouchi
Summer 2006
CHAPTER
3
Forecasting
What is Forecasting?
FORECAST:
A statement about the future value of a variable of
interest such as demand.
Forecasts affect decisions and activities throughout
an organization
Accounting, finance
Human resources
Marketing
MIS
Operations
Product / service design
Uses of Forecasts
Accounting Cost/profit estimates
Finance Cash flow and funding
Human Resources Hiring/recruiting/training
Marketing Pricing, promotion, strategy
MIS IT/IS systems, services
Operations Schedules, MRP, workloads
Product/service design New products and services
Common in all forecasts
Assumes causal system
past ==> future
Forecasts rarely perfect because of
randomness
Forecasts more accurate for
groups vs. individuals I see that you will
get an A this semester.
Forecast accuracy decreases
as time horizon increases
Elements of a Good Forecast
Timely
Reliable Accurate
Written
Steps in the Forecasting Process
“The forecast”
Step 6 Monitor the forecast
Step 5 Prepare the forecast
Step 4 Gather and analyze data
Step 3 Select a forecasting technique
Step 2 Establish a time horizon
Step 1 Determine purpose of forecast
Types of Forecasts
Judgmental - uses subjective inputs
Time series - uses historical data
assuming the future will be like the past
Associative models - uses explanatory
variables to predict the future
Judgmental Forecasts
Executive opinions
Sales force opinions
Consumer surveys
Outside opinion
Time Series Forecasts
Trend - long-term movement in data
Seasonality - short-term regular variations in
data
Cycle – wavelike variations of more than one
year’s duration
Irregular variations - caused by unusual
circumstances
Random variations - caused by chance
Forecast Variations
Irregular
variatio
n
Trend
Cycles
90
89
88
Seasonal variations
Naive Forecasts
Uh, give me a minute....
We sold 250 wheels last
week.... Now, next week
we should sell....
The forecast for any period equals
the previous period’s actual value.
Uses for Naive Forecasts
Stable time series data
F(t) = A(t-1)
Seasonal variations
F(t) = A(t-n)
Data with trends
F(t) = A(t-1) + (A(t-1) – A(t-2))
Naive Forecasts
Simple to use
Virtually no cost
Quick and easy to prepare
Easily understandable
Can be a standard for accuracy
Cannot provide high accuracy
Techniques for Averaging
Moving average
Weighted moving average
Exponential smoothing
Moving Averages
Moving average – A technique that averages a
number of recent actual values, updated as new
values become available. n
1 Ai
i=
MAn =
n
The demand for tires in a tire store in the past 5
weeks were as follows. Compute a three-period
moving average forecast for demand in week 6.
83 80 85 90 94
Moving average & Actual demand
Moving Averages
Weighted moving average – More recent values in a
series are given more weight in computing the
forecast.
Example:
For the previous demand data, compute a weighted
average forecast using a weight of .40 for the most
recent period, .30 for the next most recent, .20 for the
next and .10 for the next.
If the actual demand for week 6 is 91, forecast demand
for week 7 using the same weights.
Exponential Smoothing
Ft = Ft-1 + (At-1 - Ft-1)
• The most recent observations might have the
highest predictive value.
Therefore, we should give more weight to the
more recent time periods when forecasting.
Exponential Smoothing
Ft = Ft-1 + (At-1 - Ft-1)
Weighted averaging method based on previous
forecast plus a percentage of the forecast error
A-F is the error term, is the % feedback
Example - Exponential Smoothing
Period Actual 0.1 Error 0.4 Error
1 83
2 80 83 -3.00 83 -3
3 85 82.70 2.30 81.80 3.20
4 89 82.93 6.07 83.08 5.92
5 92 83.54 8.46 85.45 6.55
6 95 84.38 10.62 88.07 6.93
7 91 85.44 5.56 90.84 0.16
8 90 86.00 4.00 90.90 -0.90
9 88 86.40 1.60 90.54 -2.54
10 93 86.56 6.44 89.53 3.47
11 92 87.20 4.80 90.92 1.08
12 87.68 91.35
Picking a Smoothing Constant
Exponential Smoothing
Actual Alpha=0.10 Alpha=0.40
100
95
90
Demand
85
80
75
70
2 3 4 5 6 7 8 9 10 11
Period
Problem 1
National Mixer Inc. sells can openers.
Monthly sales for a seven-month period
were as follows: Month Sales
Forecast September sales volume using (1000)
each of the following: Feb 19
A five-month moving average Mar 18
Exponential smoothing with a smoothing Apr 15
constant equal to .20, assuming a March May 20
forecast of 19. Jun 18
The naive approach Jul 22
A weighted average using .60 for August, Aug 20
.30 for July, and .10 for June.
Problem 2
A dry cleaner uses exponential smoothing to
forecast equipment usage at its main plant. August
usage was forecast to be 88% of capacity. Actual
usage was 89.6%. A smoothing constant of 0.1 is
used.
Prepare a forecast for September
Assuming actual September usage of 92%, prepare
a forecast of October usage
Problem 3
An electrical contractor’s records during the last five
weeks indicate the number of job requests:
Week: 1 2 3 4 5
Requests: 20 22 18 21 22
Predict the number of requests for week 6 using each of
these methods:
Naïve
A four-period moving average
Exponential smoothing with a smoothing constant of .30.
Use 20 for week 2 forecast.
Review: forecast
Assumes causal system
past ==> future
Forecasts rarely perfect because of
randomness
Forecasts more accurate for
groups vs. individuals
Forecast accuracy decreases
as time horizon increases
Review: forecast
Naïve technique
Stable time series data
Seasonal variations
Data with trends
Averaging
Moving average
Weighted moving average
Exponential smoothing
Techniques for Trend
• Develop an equation that will suitably describe
trend, when trend is present.
• The trend component may be linear or nonlinear
• We focus on linear trends
Common Nonlinear Trends
Parabolic
Exponential
Growth
Linear Trend Equation
Ft
Ft = a + bt
Ft = Forecast for period t 0 1 2 3 4 5 t
t = Specified number of time periods
a = Value of Ft at t = 0
b = Slope of the line
Example: Ft =10+2t. Interpret 10 and 2. Plot F
Example
Sales for over the last 5 weeks are shown below:
Week: 1 2 3 4 5
Sales: 150 157 162 166 177
Plot the data and visually check to see if a linear
trend line is appropriate.
Determine the equation of the trend line
Predict sales for weeks 6 and 7.
Line chart
Sales
180
175
170
165
160
Sales
Sales
155
150
145
140
135
1 2 3 4 5
Week
Calculating a and b
n (ty) - t y
b =
n t 2 - ( t) 2
y - b t
a =
n
Linear Trend Equation Example
t y
2
Week t Sales ty
1 1 150 150
2 4 157 314
3 9 162 486
4 16 166 664
5 25 177 885
t = 15 t = 55
2
y = 812 ty = 2499
2
( t) = 225
Linear Trend Calculation
5 (2499) - 15(812) 12495 -12180
b = = = 6.3
5(55) - 225 275 -225
812 - 6.3(15)
a = = 143.5
5
y = 143.5 + 6.3t
Linear Trend plot
Actual data Linear equation
180
175
170
165
160
155
150
145
140
135
1 2 3 4 5
Recall: Problem 1
National Mixer Inc. sells can openers.
Monthly sales for a seven-month period
were as follows: Month Sales
(1000)
Plot the monthly data Feb 19
Mar 18
Forecast September sales volume using
Apr 15
a line trend equation
May 20
Which method of forecast seems least Jun 18
appropriate? Jul 22
What does use of the term sales rather Aug 20
than demand presume?
Line chart
Sales
20
0
F M A M J
J Month A S
Problem 4
A cosmetics manufacturer’s marketing department has
developed a linear trend equation that can be used to predict
annual sales of its popular Hand & Foot Cream:
Ft 80 15t
where
Ft Annual sales (1000 bottles)
t 0 corresponds to 1990
Are annual sales increasing or decreasing? By how much?
Predict annual sales for the year 2006 using the equation.
Techniques for Seasonality
Seasonality may refer to regular annual variation. There
are two models:
Additive: expressed as a quantity (e.g., 20 units), which is
added or subtracted from the series average
Multiplicative: a percentage of the average or seasonal
relative (e.g., 1.10), which is used to multiply the value of a
series to incorporate seasonality.
Additive vs. multiplicative
Example
A furniture manufacturer wants to predict quarterly demand for a
certain loveseat for periods 15 and 16, which happen to be the
second and third quarters of a particular year. The series consists
of both trend and seasonality. The trend portion of demand is
projected using the equation
Ft 124 7.5t
Quarter relatives are
Q1 1.20, Q2 1.10, Q3 0.75, Q4 0.95
Use this information to predict demand for periods 15 and 16.
Problem
A manager is using the equation below to forecast quarterly
demand for a product:
Y(t) = 6,000 + 80t
where t = 0 at Q2 of last year
Quarter relatives are Q1 = .6, Q2 = .9, Q3 = 1.3, and Q4 = 1.2.
What forecasts are appropriate for the last quarter of this year and the
first quarter of next year?
Problem
A manager of store that sells and installs hot tubs
wants to prepare a forecast for January, February
and March of 2007. Her forecasts are a
combination of trend and seasonality. She uses the
following equation to estimate the trend component
of monthly demand:
Ft 70 5t
Where t=0 is June of 2005. Seasonal relatives are
1.10 for Jan, 1.02 for Feb, and .95 for March. What
demands should she predict?
Computing seasonal relatives
120
100
80
60
40
20
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
If your data appears to have seasonality, how do you compute the
seasonal relatives?
Computing seasonal relatives
Calculate centered moving average for each
period.
Obtain the ratio of the actual value of the period
over the centered moving average.
Number of periods needed in a centered moving
average = Number of seasons involved:
Monthly data: a 12-period moving average
Quarterly data: a 4-period moving average
Example
The manager of a parking lot has computed the
number of cars per day in the lot for three weeks.
Using a seven-period centered moving average,
calculate the seasonal relatives.
Note that a seven period centered moving average
is used because there are seven days (seasons) per
week. See seasonal relatives1.xls
Problem 5
Obtain estimates of quarter relatives for these data:
Year: 1 2 3 4
Quarter: 1 2 3 4 1 2 3 4 1 2 3 4 1
Demand: 14 18 35 46 28 36 60 71 45 54 84 88 58
Problem
The manager of a restaurant believes that her
restaurant does about 10% of its business on Sunday
through Wednesday, 15% on Thursday night, 25%
on Friday night, and 20% on Saturday night.
What seasonal relatives would describe this
situation?
Note:
An alternative to deal with seasonality is to
deseasonalize data.
Deseasonalize = Remove seasonal component
from data
Gives clearer picture of the trend (nonseasonal
component)
Deseasonalize can be done by dividing each data
point by its seasonal relative.
Forecasts: review
Judgmental - uses subjective inputs
Time series - uses historical data assuming the
future will be like the past
Naïve approach
Averaging
Techniques for trend
Trend and seasonality
Associative models - uses explanatory variables to
predict the future
Associative Forecasting
Predictor variables - used to predict values of
variable interest
Regression - technique for fitting a line to a set
of points
Least squares line - minimizes sum of squared
deviations around the line
LINEAR REGRESSION
Suppose that J&T has a new product called
“AppleGlo”, which is a household cleaner. This First-Year
AppleGlo Advertising First-Year
new product has been introduced into 14 sales
Expenditures Sales
regions over the last two years. The ($ millions) ($ millions)
Advertising expenditure vs. the first year sales Region x y
are shown in the table for each region. Maine 1.8 104
New Hampshire 1.2 68
Vermont 0.4 39
The company is considering introducing Massachusetts 0.5 43
AppleGlo into two new regions, with the Connecticut 2.5 127
advertising campaign of $2.0 and $1.5 Rhode Island 2.5 134
New York 1.5 87
million.
New Jersey 1.2 77
The company would like to predict what the Pennsylvania 1.6 102
Delaware 1.0 65
expected first year sales of AppleGlo would Maryland 1.5 101
be in each region. West Virginia 0.7 46
Virginia 1.0 52
Ohio 0.8 33
LINEAR REGRESSION
160
140
120
($Millions)
100
Sales
80
60
40
20
0
0 0.5 1 1.5 2 2.5
Advertising Expenditures ($Millions)
Questions: • How to relate advertising to sales?
• What is expected first-year sales if advertising expenditure is $1M?
• How confident are you in the estimate? How good is the fit?
Correlation
The correlation coefficient is a quantitative
measure of the strength of the linear relationship
between two variables. The correlation ranges
from + 1.0 to - 1.0. A correlation of 1.0
indicates a perfect linear relationship, whereas a
correlation of 0 indicates no linear relationship.
An algebraic formula for correlation
coefficient
n xy x y
r
[n( x ) ( x) ][n( y ) ( y ) ]
2 2 2 2
Simple Linear Regression
Simple linear regression analysis
analyzes the linear relationship that exists
between two variables.
y a bx
where:
y = Value of the dependent variable
x = Value of the independent variable
a = Population’s y-intercept
b = Slope of the population regression line
Simple Linear Regression
The coefficients of the line are
n xy x y
b
n x ( x )
2 2
a
y b x
or n
a y bx
Problem 7
The manager of a seafood restaurant was Sold (y) Price (x)
asked to establish a pricing policy on 200 6.00
lobster dinners. Experimenting with 190 6.50
prices produced the following data: 188 6.75
180 7.00
Create the scatter plot and determine if 170 7.25
a linear relationship is appropriate.
162 7.50
160 8.00
Determine the correlation coefficient
and interpret it 155 8.25
156 8.50
Obtain the regression line and interpret 148 8.75
its coefficients. 140 9.00
133 9.25
Forecast Accuracy
Source of forecast errors:
Model may be inadequate
Irregular variations
Incorrect use of forecasting technique
Random variation
Key to validity is randomness
Accurate models: random errors
Invalid models: nonrandom errors
Key question: How to determine if forecasting
errors are random?
Error measures
Error - difference between actual value and predicted
value
Mean Absolute Deviation (MAD)
Average absolute error
Mean Squared Error (MSE)
Average of squared error
Mean Absolute Percent Error (MAPE)
Average absolute percent error
MAD, MSE, and MAPE
Actual forecast
MAD =
n
2
( Actual forecast)
MSE =
n -1
Actual Forecast
Actual
100
MAPE
n
Example
Period Actual Forecast (A-F) |A-F| (A-F)^2 (|A-F|/Actual)*100
1 217 215 2 2 4 0.92
2 213 216 -3 3 9 1.41
3 216 215 1 1 1 0.46
4 210 214 -4 4 16 1.90
5 213 211 2 2 4 0.94
6 219 214 5 5 25 2.28
7 216 217 -1 1 1 0.46
8 212 216 -4 4 16 1.89
-2 22 76 10.26
MAD= 2.75
MSE= 10.86
MAPE= 1.28
Controlling the Forecast
Control chart
A visual tool for monitoring forecast errors
Used to detect non-randomness in errors
Forecasting errors are in control if
All errors are within the control limits
No patterns, such as trends or cycles, are present
Controlling the forecast
Control charts
Control charts are based on the following
assumptions:
when errors are random, they are Normally
distributed around a mean of zero.
Standard deviation of error is MSE
95.5% of data in a normal distribution is within 2
standard deviation of the mean
99.7% of data in a normal distribution is within 3
standard deviation of the mean
Upper and lower control limits are often determine
via
0 2 MSE or 0 3 MSE
Example
Compute 2s control limits for
forecast errors of previous
example and determine if the
forecast is accurate.
5.41
s MSE 3.295 3.41
2s 6.59
1.41
-0.59 0 10
Errors are all between -6.59 -2.59
and +6.59 -4.59
No pattern is observed -6.59
Therefore, according to control
chart criterion, forecast is
reliable
Problem 8
Period Demand Predicted
The manager of a travel agency has
been using a seasonally adjusted 1 1 29 1 24
forecast to predict demand for 2 1 94 200
packaged tours. The actual and 3 1 56 1 50
predicted values are 4 91 94
5 85 80
6 1 32 1 40
Compute MAD, MSE, and MAPE.
7 1 26 1 28
8 1 26 1 24
Determine if the forecast is working 9 95 1 00
using a control chart with 2s limits. Use
data from the first 8 periods to develop 10 1 49 1 50
the control chart, then evaluate the 11 98 94
remaining data with the control chart. 12 85 80
13 1 37 1 40
14 1 34 1 28
Problem
Given the following demand data, prepare a naïve
forecast for periods 2 through 10. Then determine each
forecast error, and use those values to obtain 2s control
limits. If demand in the next two periods turns out to be
125 and 130, can you conclude that the forecasts are in
control?
Period 1 2 3 4 5 6 7 8 9 10
Demand 118 117 120 119 126 122 117 123 121 124
Choosing a Forecasting Technique
No single technique works in every situation
Two most important factors
Cost
Accuracy
Other factors include the availability of:
Historical data
Computers
Time needed to gather and analyze the data
Forecast horizon