12-1forecasting chapter 12 forecasting ahnaf abbas management mathematics-76

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.

Basic Assumptions

12-6 Forecasting

Elements of a Good ForecastElements of a Good Forecast

Timely

AccurateReliable

ningfu

Written

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

We’ll guess same as last month

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

We’ll guess same as last monthplus a little more for a possible trend

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

This is easy, who needs forecasting

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Continue with our successful method: guess the same as last month plus a little more for a possible trend

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Definitely looks like a trend

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Trend might be a tad steeper than I thought

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Momentary deviation, trend will continue

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

See, I told you this was easy!

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Trend will continue

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Opps, another momentary fluctuation:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Trend should continue

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Oh oh!

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Sales has leveled off:Lets average last few points

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Oh oh, maybe things are going down hill

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Let’s be conservative andAssume a negative trend

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Thank goodness, we are still basically level

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

We’ll guess same as last month

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

This stuff is easy

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

We have for sure leveled off

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Big trouble!!!Chief forecaster Joshi andCEO Joshi1 fired!

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

New chief forecaster pointsout the obvious trend

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Remarkable turnaround in sales.New CEO Joshi2 given credit

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Still looks like a trend to me

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Maybe not!

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Level except for anomaly

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Have things turned around?

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

I’ll hedge my bets

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Things have turned around.Perhaps Joshi2 truly is a genius

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Trend up!

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Not bad!

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Revise trend a tad

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Joshi2 makes cover of Fortune

Joshi1

Joshi2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

This is easy!!

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

No big deal, trend continues

(in an unrelated matterJoshi2 cashes out stock options)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Heads will surely roll soon

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Let’s be cautiously optimistic

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Joshi2 calledbefore board

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Perhaps we over reacted

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

We will guess level

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

Back to normal!

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Monthly Sales

Forecast

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

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.

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.

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

12-70Forecasting

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

12-71Forecasting

Mean Absolute Deviation - 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.

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

1 2 3 4 5 6 7 8 9 10 11 12

Actual

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

MSE = Actual forecast)

MAPE = Actual forecas

/ 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

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

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

1 2 3 4 5 6 7 8 9 10 11 12

Period

and .1

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.

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.

12-89Forecasting

ExercisesExercises

12-90Forecasting

ExercisesExercises

12-91Forecasting

ExercisesExercises

12-1forecasting chapter 12 forecasting ahnaf abbas management mathematics-76

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