12-1forecasting chapter 12 forecasting ahnaf abbas management mathematics-76
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12-1 Forecasting
CHAPTER12
Forecasting
AHNAF ABBASManagement Mathematics-76.
12-2 Forecasting
CHAPTER12
Objectives
AHNAF ABBASManagement Mathematics-76.
1.How to Classify Forecasts2.How to Calculate Moving Averages3.How to Perform Exponential
Smoothing.
12-3 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
12-4 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, workloads
Product/service design New products and services
Uses of ForecastsUses of Forecasts
12-5 Forecasting
Assumes causal systempast ==> future
Past patterns (behavior) will continue into future.
Forecasts rarely perfect because of randomness
Forecast accuracy decreases as time horizon increases
I see that you willget an A this semester.
JOSHI
Basic Assumptions
12-6 Forecasting
Elements of a Good ForecastElements of a Good Forecast
Timely
AccurateReliable
Mea
ningfu
l
Written
Easy
to u
se
12-7 Forecasting
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”
12-8 Forecasting
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.
12-9 Forecasting
Simple to use Virtually no cost Quick and easy to prepare Data analysis is nonexistent Easily understandable Cannot provide high accuracy
Naïve ForecastsNaïve Forecasts
12-10Forecasting
Types of ForecastsTypes of Forecasts
Types By lead time
The time between when the forecast is made and the future point to which it refers.
Long-term: more than 10 years. Medium-term: up to 5 years. short-term: months to a year.
12-11Forecasting
Types of ForecastsTypes of Forecasts
Univariate :Using past patterns
e.g. Time series which uses historical data assuming the future will be like the past.
Multivariate- uses explanatory variables to predict the future, i.e. Past relationships between multiple variables.
Qualitative Judgmental - uses subjective inputs
12-12Forecasting
Judgmental Forecasts (Qualitative)Judgmental Forecasts (Qualitative)
Executive opinions Sales force opinions Consumer surveys Outside opinion Delphi method
Opinions of managers and staff
12-13Forecasting
Time Series ForecastsTime Series Forecasts
A time series is a continuous set of observations that are ordered in equally spaced intervals(e.g one per month).
Basic concept of univariate forecasting:
Future values = f( Past values )
e.g. Two months average sales :
Forecast for June =
(April sales + May sales ) / 2
Univariate methods includes smoothing (averages) and exponential smoothing.
12-14Forecasting
Multivariate ForecastsMultivariate Forecasts
Known as Causal methods: make projections of the future by modeling the relationship between a series and other series.
e.g. A forecast for furniture sales may be based on a relationship between economic indicators such as housing starts, personal income ,No. of new marriages etc… :
Future values = f( Past values, Values of other variables )e.g. June demand = 50 + 0.2 MS + 1xAPI +0.5NHMultivariate methods include multiple regression and
econometric.
12-15ForecastingMonthly Sales and Forecast
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We’ll guess same as last month
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We’ll guess same as last monthplus a little more for a possible trend
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This is easy, who needs forecasting
12-20ForecastingMonthly Sales and Forecast
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Continue with our successful method: guess the same as last month plus a little more for a possible trend
12-21ForecastingMonthly Sales and Forecast
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Definitely looks like a trend
12-23ForecastingMonthly Sales and Forecast
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Forecast
Trend might be a tad steeper than I thought
12-25ForecastingMonthly Sales and Forecast
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Opps
12-26ForecastingMonthly Sales and Forecast
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Momentary deviation, trend will continue
12-27ForecastingMonthly Sales and Forecast
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See, I told you this was easy!
12-28ForecastingMonthly Sales and Forecast
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Forecast
Trend will continue
12-29ForecastingMonthly Sales and Forecast
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Forecast
Opps, another momentary fluctuation:
12-30ForecastingMonthly Sales and Forecast
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Forecast
Trend should continue
12-31ForecastingMonthly Sales and Forecast
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Oh oh!
12-32ForecastingMonthly Sales and Forecast
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Sales has leveled off:Lets average last few points
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Oh oh, maybe things are going down hill
12-34ForecastingMonthly Sales and Forecast
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Let’s be conservative andAssume a negative trend
12-35ForecastingMonthly Sales and Forecast
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Thank goodness, we are still basically level
12-36ForecastingMonthly Sales and Forecast
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We’ll guess same as last month
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This stuff is easy
12-38ForecastingMonthly Sales and Forecast
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We have for sure leveled off
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Big trouble!!!Chief forecaster Joshi andCEO Joshi1 fired!
12-40ForecastingMonthly Sales and Forecast
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New chief forecaster pointsout the obvious trend
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Remarkable turnaround in sales.New CEO Joshi2 given credit
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Still looks like a trend to me
12-43ForecastingMonthly Sales and Forecast
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Maybe not!
12-44ForecastingMonthly Sales and Forecast
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Level except for anomaly
12-45ForecastingMonthly Sales and Forecast
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Forecast
Have things turned around?
12-46ForecastingMonthly Sales and Forecast
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I’ll hedge my bets
12-47ForecastingMonthly Sales and Forecast
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Things have turned around.Perhaps Joshi2 truly is a genius
12-48ForecastingMonthly Sales and Forecast
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Trend up!
12-49ForecastingMonthly Sales and Forecast
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Not bad!
12-50ForecastingMonthly Sales and Forecast
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Forecast
Revise trend a tad
12-51ForecastingMonthly Sales and Forecast
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Joshi2 makes cover of Fortune
Joshi1
Joshi2
12-52ForecastingMonthly Sales and Forecast
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This is easy!!
12-53ForecastingMonthly Sales and Forecast
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Forecast
No big deal, trend continues
(in an unrelated matterJoshi2 cashes out stock options)
12-54ForecastingMonthly Sales and Forecast
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Forecast
12-55ForecastingMonthly Sales and Forecast
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Forecast
Heads will surely roll soon
12-56ForecastingMonthly Sales and Forecast
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Forecast
Let’s be cautiously optimistic
12-57ForecastingMonthly Sales and Forecast
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Forecast
Joshi2 calledbefore board
12-58ForecastingMonthly Sales and Forecast
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Forecast
12-59ForecastingMonthly Sales and Forecast
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Forecast
Perhaps we over reacted
12-60ForecastingMonthly Sales and Forecast
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We will guess level
12-61ForecastingMonthly Sales and Forecast
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Back to normal!
12-62ForecastingMonthly Sales and Forecast
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Forecast
12-63ForecastingMonthly Sales and Forecast
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Joshi2 fired!
12-64Forecasting
Structure of a Time SeriesStructure of a Time Series
Trend patterns- general increase or decrease in a time series that lasts for certain period.
population changes,changes in economic conditions..
Seasonality – results from the events that are periodic.Common seasonal influences :
climate ,human habits ,holidays,…. Cycle – wavelike variations of more than one year’s
duration. economic and business expansions ,contractions. Random variations - caused by chance,many
influences that act independently to yield non repeating patterns.
12-65Forecasting
Forecast VariationsForecast Variations
Trend
Irregularvariation
Seasonal variations
908988
Figure 3.1
Cycles
12-66Forecasting
Importance of Pattern :
Actual value = Pattern + Error Predicting values using Mean, Median or Mode
Mean : arithmetic average ,measures that value about which 50% of deviations are above and 50% are below.i.e. Sum(Deviations) = 0
e.g. Time t = 1 2 3 4 5 6 7 (seven months)
Value X=11 4 5 12 9 2 6; X = sales
Mean Sales = 7
Sum(Deviations) = 0
Statistical Fundamentals 4 Forecasting Statistical Fundamentals 4 Forecasting
12-67Forecasting
The Median : 50% of values are above and 50% are below.
e.g. 2 4 5 6 9 11 12
Median = 6
When an even number of observations exists, Median = mean of middle two values.
The Mode : is the number or group of numbers that occur most often.
Statistical Fundamentals 4 Forecasting Statistical Fundamentals 4 Forecasting
12-68Forecasting
Comparison of Measures: The Mean : Because it is the center of deviations, the
mean is greatly influenced by the
extreme values.
e.g. A very large deviation (Outlier) will greatly affect the value of the mean. It is not appropriate without adjusting the outlier.
The Median : is not affected by extreme values.
Useful in describing highly skewed data.
The Mode: is not affected by extremes.
Statistical Fundamentals 4 Forecasting Statistical Fundamentals 4 Forecasting
12-69Forecasting
Example: Consider this sales time series for forecasting Period 9 : Period t = 1 2 3 4 5 6 7 8Value X = 100 70 90 110 1200 110 130 80Mean = 236.25 ; Median = 105 ; Mode = 110.Best estimate = 130 ;
Mean = 102.5 ; Median = 105 ; Mode = 110 &130
Statistical Fundamentals 4 Forecasting Statistical Fundamentals 4 Forecasting
12-70Forecasting
Statistical Fundamentals 4 Forecasting Statistical Fundamentals 4 Forecasting
Measuring Errors-Standard deviation & MAD
Standard Deviation: The square root of the mean of the squared deviations.
For a population : (RMS)
Known as Root Mean Square Error
For a Sample: N
X t
2)(
1
)( 2
n
XXS t
12-71Forecasting
Statistical Fundamentals 4 Forecasting Statistical Fundamentals 4 Forecasting
Mean Absolute Deviation - MAD
MAD =
Comparison of Error Measures:
MAD and Standard deviation are equivalent measures of dispersion.
Despite the popularity of MAD , standard deviation is the preferred measure in forecasting.The more high the RMS& MAD ,the better the Forecast.
n
XX t
12-72Forecasting
Univariate MethodsUnivariate Methods
Moving average
Weighted moving average
Exponential smoothing
Are effective methods for short term forecasts
(series that do not have cyclical or seasonal patterns)
12-73Forecasting
Univariate Methods-Moving AveragesUnivariate Methods-Moving Averages
Moving averages: Future value will equal an average of past values.
Useful in random series because it averages or smoothes the most recent actual values to remove the unwanted randomness.
Remember :
Forecast error = Actual - Forecast
12-74Forecasting
Univariate Methods-Moving AveragesUnivariate Methods-Moving Averages
Example:
Month t Actual MA2 Error MA4 Error
Jan 1 120Feb 2 124Mar 3 122 122 0.00April 4 123 123 0.00May 5 125 122.5 2.50 122.25 2.75 Jun 6 128 124 4.00 123.5 4.50Jul 7 129 126.5 2.5 124.5 4.50Aug 8 127 128.5 -1.5 126.5 0.75
12-75Forecasting
Techniques for AveragingTechniques for Averaging
1. Each new forecast moves ahead one period by adding the newest actual and dropping the oldest actual.
MA4(May) = (Jan+Feb+Mar+Apr)/4
MA4(Jun) = (Feb+Mar+Apr+May)/4
2. Response to Recent data:
MA’s Forecasts can be made less responsive to recent data by increasing the number of past observations.
MA’s Forecasts can be made more responsive to recent data by decreasing the number of past observations.
12-76Forecasting
Simple Moving AverageSimple Moving Average
MAn = n
Aii = 1n
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Actual
MA3
MA5
12-77Forecasting
Weighted Moving AveragesWeighted Moving Averages
Weight 0.10 0.20 0.30 0.40 Month 1 2 3 4_ 5 Sales 100 90 105 95 ?
F5 = 0.40(95) + 0.30(105) + 0.20(90) + 0.10(100) = 97.5 units
12-78Forecasting
Exponential SmoothingExponential Smoothing
Smoothing constant determines the weight given to the most recent observations, it controls the rate of smoothing or averaging.
11 )1( ttt FAF Ft = Exponentially smoothed forecast for period t.
At -1= Actual in the prior period.
Ft -1 = Exponentially smoothed forecast of the prior period
= Smoothing constant. ( 0 < < 1)
12-79Forecasting
Exponential SmoothingExponential Smoothing
Example: suppose a company desires to forecast demand for a product with = 0.3 .
Last month’s actual = 1000 and the forecast was
900, the forecast for this month:
Ft = (0.3)(1000) + (1-0.3)(900) = 930
Alpha provides the relative weight given to each term of the equation. With alpha = 0.3 ,the forecast is 30% of the most recent Actual and 70% of the most recent Forecasted value.
12-80Forecasting
Forecast AccuracyForecast Accuracy The fitted region is where we assume the data is known and
we fit the model to this data. The forecast region is where we assume the data is not known
and has to be forecast using the model derived from the fitted region.
Mean Absolute Deviation (MAD) Average absolute error
Mean Squared Error (MSE) Average of squared error
Mean Absolute Percent Error (MAPE) Average absolute percent error
12-81Forecasting
MAD, MSE, and MAPEMAD, MSE, and MAPE
MAD = Actual forecast
n
MSE = Actual forecast)
2
n-1
(
MAPE = Actual forecas
t
n
/ Actual*100)
12-82Forecasting
Absolute Error MeasuresAbsolute Error Measures
et = At - Ft Month Actual Forec. Error. |Error| Error2 St. Dev. Jan. 10 15 -5 5 25 6.25 Feb. 12 14 -2 2 4 4.38 Mar. 14 13.6 0.4 0.4 0.16 3.09
Apr. 13 13.68 -0.68 0.68 0.46 2.53 49 56.28 -7.28 8.08 29.62
ME : mean error = = -1.82; MAD = = 0.5
SSE : sum of square errors = = 29.62
n
et n
et
2te
12-83Forecasting
Exponential SmoothingExponential SmoothingSteps in Using Exponential smoothing:
1. Choice of a smoothing constant:The best alphas should be chosen on the basis of Minimal sum of squared errors.
2. An initial forecast: the first actual value is chosen as the forecast for the second period.
Example : Period Actual Forecast =0.3
1 900
2 1000 900
F3 = A2 +(1 - )F2 = (0.3)(1000) + (1 -0.3)(900)=930
12-84Forecasting
Period Actual Alpha = 0.1 Error Alpha = 0.4 Error1 422 40 42 -2.00 42 -23 43 41.8 1.20 41.2 1.84 40 41.92 -1.92 41.92 -1.925 41 41.73 -0.73 41.15 -0.156 39 41.66 -2.66 41.09 -2.097 46 41.39 4.61 40.25 5.758 44 41.85 2.15 42.55 1.459 45 42.07 2.93 43.13 1.87
10 38 42.36 -4.36 43.88 -5.8811 40 41.92 -1.92 41.53 -1.5312 41.73 40.92
Example - Exponential SmoothingExample - Exponential Smoothing
12-85Forecasting
Picking a Smoothing ConstantPicking a Smoothing Constant
35
40
45
50
1 2 3 4 5 6 7 8 9 10 11 12
Period
Dem
and .1
.4
Actual
12-86Forecasting
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
12-87Forecasting
The Smoothing ConstantThe Smoothing Constant
1. The weights of the past Actuals are determined by alpha:
Most recent :
One period old :
Two periods old : etc…
2. The alpha that yields the most accurate forecast
is the one that achieves the lowest MSE.
1)1(
2)1(
12-88Forecasting
The Smoothing ConstantThe Smoothing Constant3. If = 1 ,then Ft = At – 1 (zero smoothing)
4. Relation with number of periods :
e.g. N = 19 , then
Useful when converting from or to MA’s
Response to recent data:
More responsive : higher alpha.
Less responsive : lower alpha.
1
2
N
1.0
12-89Forecasting
ExercisesExercises
12-90Forecasting
ExercisesExercises
12-91Forecasting
ExercisesExercises