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Transcript of forecasting
College of Business AdministrationCal State San Marcos
Production & Operations ManagementHTM 305
Dr. M. Oskoorouchi
Summer 2006
CHAPTER3
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
What is Forecasting?What is Forecasting?
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
Uses of ForecastsUses of Forecasts
Assumes causal systempast ==> future
Forecasts rarely perfect because of randomness
Forecasts more accurate forgroups vs. individuals
Forecast accuracy decreases as time horizon increases I see that you will
get an A this semester.
Common in all forecastsCommon in all forecasts
Elements of a Good ForecastElements of a Good Forecast
Timely
AccurateReliable
Mea
ningfu
l
Written
Easy
to u
se
Steps in the Forecasting ProcessSteps in the Forecasting Process
Step 1 Determine purpose of forecast
Step 2 Establish a time horizon
Step 3 Select a forecasting technique
Step 4 Gather and analyze data
Step 5 Prepare the forecast
Step 6 Monitor the forecast
“The forecast”
Types of ForecastsTypes 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 ForecastsJudgmental Forecasts
Executive opinions
Sales force opinions
Consumer surveys
Outside opinion
Time Series ForecastsTime 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 VariationsForecast Variations
Trend
Irregularvariation
Seasonal variations
908988
Cycles
Naive ForecastsNaive Forecasts
Uh, give me a minute.... We sold 250 wheels lastweek.... Now, next week we should sell....
The forecast for any period equals the previous period’s actual value.
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))
Uses for Naive ForecastsUses for 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
Naive ForecastsNaive Forecasts
Techniques for AveragingTechniques for Averaging
Moving average
Weighted moving average
Exponential smoothing
Moving AveragesMoving Averages
Moving average – A technique that averages a number of recent actual values, updated as new values become available.
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
MAn = n
Aii = 1n
Moving average & Actual demandMoving average & Actual demand
Moving AveragesMoving 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 SmoothingExponential Smoothing
• The most recent observations might have the highest predictive value.
Therefore, we should give more weight to the more recent time periods when forecasting.
Ft = Ft-1 + (At-1 - Ft-1)
Exponential SmoothingExponential Smoothing
Weighted averaging method based on previous forecast plus a percentage of the forecast error
A-F is the error term, is the % feedback
Ft = Ft-1 + (At-1 - Ft-1)
Example - Exponential SmoothingExample - Exponential Smoothing
Period Actual 0.1 Error 0.4 Error1 832 80 83 -3.00 83 -33 85 82.70 2.30 81.80 3.204 89 82.93 6.07 83.08 5.925 92 83.54 8.46 85.45 6.556 95 84.38 10.62 88.07 6.937 91 85.44 5.56 90.84 0.168 90 86.00 4.00 90.90 -0.909 88 86.40 1.60 90.54 -2.54
10 93 86.56 6.44 89.53 3.4711 92 87.20 4.80 90.92 1.0812 87.68 91.35
Picking a Smoothing ConstantPicking a Smoothing Constant
Exponential Smoothing
70
75
80
85
90
95
100
2 3 4 5 6 7 8 9 10 11
Period
Dem
and
Actual Alpha=0.10 Alpha=0.40
Problem 1 Problem 1
National Mixer Inc. sells can openers. Monthly sales for a seven-month period were as follows: Forecast September sales volume using
each of the following: A five-month moving average Exponential smoothing with a smoothing
constant equal to .20, assuming a March forecast of 19.
The naive approach A weighted average using .60 for August,
.30 for July, and .10 for June.
Month Sales
(1000)
Feb 19
Mar 18
Apr 15
May 20
Jun 18
Jul 22
Aug 20
Problem 2 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 Problem 3 An electrical contractor’s records during the last five
weeks indicate the number of job requests:Week: 1 2 3 4 5Requests: 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.
Assumes causal systempast ==> future
Forecasts rarely perfect because of randomness
Forecasts more accurate forgroups vs. individuals
Forecast accuracy decreases as time horizon increases
Review: forecastReview: forecast
Review: forecastReview: forecast
Naïve technique Stable time series data Seasonal variations Data with trends
Averaging Moving average Weighted moving average Exponential smoothing
Techniques for TrendTechniques 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 TrendsCommon Nonlinear Trends
Parabolic
Exponential
Growth
Linear Trend EquationLinear Trend Equation
Ft = Forecast for period 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
Ft = a + bt
0 1 2 3 4 5 t
Ft
ExampleExample
Sales for over the last 5 weeks are shown below:
Week: 1 2 3 4 5Sales: 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 chartLine chart
Sales
135
140
145
150
155
160
165
170
175
180
1 2 3 4 5
Week
Sale
s
Sales
Calculating a and bCalculating a and b
b = n (ty) - t y
n t2 - ( t)2
a = y - b t
n
Linear Trend Equation ExampleLinear Trend Equation Example
t yW e e k t 2 S a l e s t y
1 1 1 5 0 1 5 02 4 1 5 7 3 1 43 9 1 6 2 4 8 64 1 6 1 6 6 6 6 45 2 5 1 7 7 8 8 5
t = 1 5 t 2 = 5 5 y = 8 1 2 t y = 2 4 9 9( t ) 2 = 2 2 5
Linear Trend CalculationLinear Trend Calculation
y = 143.5 + 6.3t
a = 812 - 6.3(15)
5 =
b = 5 (2499) - 15(812)
5(55) - 225 =
12495-12180
275 -225 = 6.3
143.5
Linear Trend plotLinear Trend plot
135
140
145
150
155
160
165
170
175
180
1 2 3 4 5
Actual data Linear equation
Recall: Problem 1 Recall: Problem 1
National Mixer Inc. sells can openers. Monthly sales for a seven-month period were as follows:
Plot the monthly data Forecast September sales volume using
a line trend equation Which method of forecast seems least
appropriate? What does use of the term sales rather
than demand presume?
Month Sales
(1000)
Feb 19
Mar 18
Apr 15
May 20
Jun 18
Jul 22
Aug 20
Line chartLine chart
Month
Sales
F M A M JJ A S
20
0
Problem 4Problem 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:
Are annual sales increasing or decreasing? By how much? Predict annual sales for the year 2006 using the equation.
80 15
Annual sales (1000 bottles)
0 corresponds to 1990
t
t
F t
where
F
t
Techniques for SeasonalityTechniques 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. multiplicativeAdditive vs. multiplicative
Example 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
Quarter relatives are
Use this information to predict demand for periods 15 and 16.
124 7.5tF t
1 2 3 41.20, 1.10, 0.75, 0.95Q Q Q Q
Problem 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?
ProblemProblem 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:
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?
70 5tF t
Computing seasonal relativesComputing seasonal relatives
0
20
40
60
80
100
120
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 relativesComputing 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 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 5Problem 5
Obtain estimates of quarter relatives for these data:
Year: 1 2 3 4
Quarter:
Demand:
1 2 3 4
14 18 35 46
1 2 3 4
28 36 60 71
1 2 3 4
45 54 84 88
1
58
Problem 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: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: reviewForecasts: 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 ForecastingAssociative 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
AppleGloFirst-Year
AdvertisingExpenditures($ millions)
First-YearSales
($ millions)Region x y
Maine 1.8 104New Hampshire 1.2 68Vermont 0.4 39Massachusetts 0.5 43Connecticut 2.5 127Rhode Island 2.5 134New York 1.5 87New Jersey 1.2 77Pennsylvania 1.6 102Delaware 1.0 65Maryland 1.5 101West Virginia 0.7 46Virginia 1.0 52Ohio 0.8 33
Suppose that J&T has a new product called Suppose that J&T has a new product called “AppleGlo”, which is a household cleaner. “AppleGlo”, which is a household cleaner. This new product has been introduced into 14 This new product has been introduced into 14 sales regions over the last two years. The sales regions over the last two years. The Advertising expenditure vs. the first year sales Advertising expenditure vs. the first year sales are shown in the table for each region.are shown in the table for each region.
The company is considering introducing The company is considering introducing AppleGlo into two new regions, with the AppleGlo into two new regions, with the advertising campaign of $2.0 and $1.5 advertising campaign of $2.0 and $1.5 million.million.
The company would like to predict what the The company would like to predict what the expected first year sales of AppleGlo would expected first year sales of AppleGlo would be in each region.be in each region.
LINEAR REGRESSIONLINEAR REGRESSIONLINEAR REGRESSIONLINEAR REGRESSION
Questions:Questions: • • How to relate advertising to sales? How to relate advertising to sales? •• What is expected first-year sales if advertising expenditure is $1M?What is expected first-year sales if advertising expenditure is $1M? •• How confident are you in the estimate? How good is the fit?How confident are you in the estimate? How good is the fit?
LINEAR REGRESSIONLINEAR REGRESSIONLINEAR REGRESSIONLINEAR REGRESSION
0
20
40
60
80
100
120
140
160
0 0.5 1 1.5 2 2.5
Advertising Expenditures ($Millions)
S
ales
($
Mil
lion
s)
CorrelationCorrelation
The correlation coefficientcorrelation 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 An algebraic formula for correlation coefficientcoefficient
])()(][)()([ 2222 yynxxn
yxxynr
Simple Linear RegressionSimple Linear Regression
Simple linear regression analysisSimple linear regression analysis analyzes the linear relationship that exists
between two variables.
bxay where:
y = Value of the dependent variablex = Value of the independent variablea = Population’s y-interceptb = Slope of the population regression line
Simple Linear RegressionSimple Linear Regression
The coefficients of the line are
or
22 )( xxn
yxxynb
xbya
y b xa
n
Problem 7Problem 7
The manager of a seafood restaurant was asked to establish a pricing policy on lobster dinners. Experimenting with prices produced the following data:
Create the scatter plot and determine if a linear relationship is appropriate.
Determine the correlation coefficient and interpret it
Obtain the regression line and interpret its coefficients.
Sold (y) Price (x)
200 6.00
190 6.50
188 6.75
180 7.00
170 7.25
162 7.50
160 8.00
155 8.25
156 8.50
148 8.75
140 9.00
133 9.25
Forecast AccuracyForecast 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 measuresError 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 MAPEMAD, MSE, and MAPE
MAD = Actual forecast
n
MSE = Actual forecast)
-1
2
n
(
Actual Forecast100
ActualMAPEn
ExampleExample
Period Actual Forecast (A-F) |A-F| (A-F)^2 (|A-F|/Actual)*1001 217 215 2 2 4 0.922 213 216 -3 3 9 1.413 216 215 1 1 1 0.464 210 214 -4 4 16 1.905 213 211 2 2 4 0.946 219 214 5 5 25 2.287 216 217 -1 1 1 0.468 212 216 -4 4 16 1.89
-2 22 76 10.26
MAD= 2.75MSE= 10.86
MAPE= 1.28
Controlling the ForecastControlling 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 forecastControlling the forecast
Control chartsControl 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 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
MSE
0 2 0 3MSE or MSE
Example Example Compute 2s control limits for
forecast errors of previous example and determine if the forecast is accurate.
Errors are all between -6.59 and +6.59
No pattern is observed Therefore, according to control
chart criterion, forecast is reliable
-6.59
-4.59
-2.59
-0.59
1.41
3.41
5.41
0 10
3.295
2 6.59
s MSE
s
Problem 8Problem 8
The manager of a travel agency has been using a seasonally adjusted forecast to predict demand for packaged tours. The actual and predicted values are
Compute MAD, MSE, and MAPE.
Determine if the forecast is working using a control chart with 2s limits. Use data from the first 8 periods to develop the control chart, then evaluate the remaining data with the control chart.
Period Demand Predicted
1 129 1242 194 2003 156 1504 91 945 85 806 132 1407 126 1288 126 1249 95 100
10 149 15011 98 9412 85 8013 137 14014 134 128
ProblemProblem
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 TechniqueChoosing 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