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