Ch6 Forecasting

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OPIM201 BUSINESS PROCESSES

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Chapter 6: Forecasting

1. Introduction

Forecasting is vital to every business organization and for every significant

management decision. Forecasting is the basis of corporate long-run planning. Inthe functional areas of finance and accounting, forecasts provide the basis forbudgetary planning and cost control. Marketing relies on sales forecasting to plannew products, compensate sales personnel, and make other key decisions.Production and operations personnel use forecasts to make periodic decisionsinvolving process selection, capacity planning, and facility layout, as well as forcontinual decisions about production planning scheduling, and inventory.

Bear in mind that a perfect forecast is usually impossible. Too many factors in thebusiness environment cannot be predicted with certainty. Therefore, rather thansearch for the perfect forecast, it is far more important to establish the practice of

continual review of forecasts and to learn to live with inaccurate forecasts. Whenforecasting, a good strategy is to use two or three methods and look at them for thecommonsense view.

2. Types of forecasting

Forecasting can be classified into four basic types:Qualitative: Qualitative techniques are subjective or judgmental and are based onestimates and opinions.Time series analysis: This is based on the idea that data relating to past demandcan be used to predict future demand.

Causal relationships: Causal forecasting, which we discuss using the linearregression technique, assumes that demand is related to some underlying factor orfactors in the environment.Simulation: Simulation models allow the forecaster to run through a range ofassumptions about the condition of the forecast.

3. Qualitative techniques in forecasting

Grass roots: this method builds the forecast by adding successively from thebottom. The assumption here is that the person closest to the customer or end userof the product knows its future needs best. Forecasts at this bottom level are

summed and given to the next higher level. This is usually a district warehouse,which then adds in safety stocks and any effects of ordering quantity sizes. Thisamount is then fed to the next level, which may be a regional warehouse. Theprocedure repeats until it becomes an input at the top level, which, in the case of amanufacturing firm, would be the input to the production system.

Market research: Firms often hire outside companies that specialize in marketresearch to conduct this type of forecasting. Market research is used mostly forproduct research in the sense of looking for new product ideas, likes and dislikes


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about existing products, which competitive products within a particular class arepreferred, and so on. The data collection methods are primarily surveys andinterviews.

Historical analogy: In trying to forecast demand for a new product, an ideal situation

would be where an existing product or generic product could be used as a model.There are many ways to classify such analogies--for example, complementaryproducts, substitutable or competitive products, and products as a function ofincome. If you buy a CD through the mail, you will receive more mail about new CDsand CD players. Another example would be toasters and coffee pots. A firm thatalready produces toasters and wants to produce coffee pots could use the toasterhistory as a likely growth model.

Panel consensus: In a panel consensus, the idea that two heads are better than oneis extrapolated to the idea that a panel of people from a variety of positions candevelop a more reliable forecast than a narrower group. Panel forecasts are

developed through open meetings with free exchange of ideas from all levels ofmanagement and individuals. The difficulty with this open style is that loweremployee levels are intimidated by higher levels of management. For example, asalesperson in a particular product line may have a good estimate of future productdemand but may not speak up to refute a much different estimate given by the vicepresident of marketing. The Delphi technique overcomes this problem.

Delphi method: this method conceals the identity of the individuals participating inthe study. Everyone has the same weight. Procedurally, a moderator creates aquestionnaire and distributes it to participants. Their responses are summed andgiven back to the entire group along with a new set of questions. This technique canusually achieve satisfactory results in three rounds.

4. Time series analysis

Time series is just a fancy term for a collection of observations of some economic orphysical phenomenon drawn at discrete points in time, usually equally spaced. Theidea is that information can be inferred from the pattern of past observations and canbe used to forecast future values of the series.

4.1. Components of demand

In most cases, demand for products or services can be broken down into sixcomponents: average demand for the period, a trend, seasonal element, cyclicalelements, random variation, and autocorrelation. See appendix for an examplefigure of these components.

Cyclical factors are more difficult to determine because the time span may beunknown or the cause of the cycle may not be considered. Cyclical influence ondemand may come from such occurrences as political elections, war, economic


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conditions, or sociological pressures. The cyclic variation is similar to seasonality,except that the length and the magnitude of the cycle may vary.

Random variations are caused by chance events. Statistically, when all the knowncauses for demand (average, trend, seasonal, cyclical, and auto-correlative) are

subtracted from total demand, what remains is the unexplained portion of demand.If we cannot identify the cause of this remainder, it is assumed to be purely randomchance.

Autocorrelation denotes the persistence of occurrence. More specifically, itmeasures the degree of dependency among values of observed data separated bya fixed number of periods.

4.2. Methods for forecasting stationary series

A stationary time series is one in which each observation can be represented by aconstant plus a random fluctuation.

4.2.1. Simple moving average

When demand for a product is neither growing nor declining rapidly, and if it doesnot have seasonal characteristics, a moving average can be useful in removing therandom fluctuations for forecasting.

In a N-period simple moving average, we take the average of last N periods as ourforecast for the next period.

Example 1:

Week ActualDemand 3-week 9-week

1 800

2 1400

3 1000

4 1500

5 1500

6 1300

7 18008 1700

9 1300

10 1700

11 1700

12 1500

13 2300

14 2300

15 2000


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Although it is important to select the best period for the moving average, there areseveral conflicting effects of different period lengths. The longer the moving averageperiod, the more the random elements are smoothed. But if there is a trend in thedata – either increasing or decreasing – the moving average has the adversecharacteristic of lagging the trend. Therefore, for a shorter time span, there is a

closer following of the trend. Conversely, a longer time span gives a smootherresponse but lags the trend.

The main disadvantage with this method is that all individual elements must becarried as data because a new forecast period involves adding new data anddropping the earliest data. The amount of data involved is significant.

Another shortcoming of this method is it lags behind the trend. Consider a demandprocess in which there is a definite trend as follows.

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

Actualdemand 2 4 6 8 10 12 14 16 18 20 22 24

3-period avg 4 6 8 10 12 14 16 18 20

6-period avg 7 9 11 13 15 17

Notice that both the 3-period and 6-period moving average forecasts lag behind thetrend, and that the forecast with a smaller N value follows the actual demand moreclosely.

4.2.2. Weighted moving average

Whereas the simple moving average gives equal weight to each component of themoving average database, a weighted moving average allows any weights to beplaced on each element, providing, of course, that the sum of all weights equals 1.For example, a department store may find that in a four-month period, the bestforecast is derived by using 40 percent of the actual sales for the most recentmonth, 30 percent of two months ago, 20 percent of three months ago, and 10percent of four months ago. If actual sales experience was

month 1 month 2 month 3 month 4 month 5

100 90 105 95 ?

The forecast for month 5 would be

F5 =

Suppose sales for month 5 actually turned out to be 110. Then the forecast formonth 6 would be

F6 =


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Choosing weights: experience and trial and error are the simplest ways to chooseweights. As a general rule, the most recent past is the most important indicator ofwhat to expect in the future, and therefore, it should get higher weighting. However,if the data are seasonal, for example, weights should be established accordingly.Bathing suit sales in July of last year should be weighted more heavily than bathing

suit sales in December.

Example 2:In Atlanta, the number of daily calls for repair of Speedy copy machines in 8 dayshas been recorded as follows:

dayactualdemand

3-day movingavg

Forecasterror

weighted movingavg

Forecasterror

1 92

2 127

3 103

4 165

5 132

6 111

7 174

8 94

a. Prepare a 3-period moving average forecast. What is the error on each day?b. Prepare a 3-period weighted moving average forecast with w

1=0.2, w

2=0.3, and

w3=0.5(most recent data carries heaviest weight). What is the error on eachday?

c. Which of the two forecasts is better? (use MAD to judge)


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4.2.3. Exponential smoothing

Exponential smoothing is the most used of all forecasting techniques. It is an integralpart of virtually all computerized forecasting programs, and it is widely used inordering inventory in retail firms, wholesale companies, and service agencies.

Exponential smoothing methods have become well accepted for six major reasons: Exponential models are surprisingly accurate Formulating an exponential model is relatively easy The user can understand how the model works Little computation is required to use the model Computer storage requirements are small because of the limited use of

historical data Tests for accuracy as to how well the model is performing are easy to

compute

In the exponential smoothing method, only three pieces of data are needed toforecast the future: the most recent forecast, the actual demand that occurred forthat forecast period, and a smoothing constant alpha ( α ). This smoothing constantdetermines the level of smoothing and the speed of reaction to differences betweenforecasts and actual occurrences. The value for the constant is determined both bythe nature of the product and by the manager’s sense of what constitutes a goodresponse rate. For example, if a firm produced a standard item with relatively stabledemand, the reaction rate to differences between actual and forecast demand wouldtend to be small, perhaps just 5 or 10 percentage points. However, if the firm wereexperiencing growth, it would be desirable to have a higher reaction rate, perhaps 15to 30 percentage points, to give greater importance to recent growth experience.The more rapid the growth, the higher the reaction rate should be.

The equation for a single exponential smoothing forecast is simply

Ft = Ft-1 + α (At-1 – Ft-1)

Example 3: Assume last month’s forecast was 1050, and 1000 actually were demanded. What isthe forecast for this month? Use α = 0.05.


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The following is some supplementary info for you to understand why we use theterm “exponential”. You can skip it if you are not interested.

Notice that Ft-1 = Ft-2 + α (At-2 – Ft-2), substituting into the above equation yields

Ft = α At-1 + α (1- α) At-2 + (1- α )2

Ft-2

We can substitute for Ft-2 in the same fashion. If we continue in this way we obtainthe infinite expansion for Ft:

1

0

)1( −−

∞

=∑ −= it i

i

t AF α α

Hence, exponential smoothing applies a declining set of weights to all past data. Infact, we can fit the continuous exponential curve α exp(-αi) to these weights, which iswhy the method is called exponential smoothing. The smoothing constant α plays

essentially the same role here as the value of N does in moving averages. If α islarge, more weight is placed on the current observation of demand and less weighton past observations, which results in forecasts that will react quickly to changes inthe demand pattern but may have much greater variation from period to period.


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4.3. Trend-based methods

Both moving average and exponential smoothing forecasts will lag behind a trend ifone exists. In this section we consider two forecasting methods that specificallyaccount for a trend in the data: enhanced exponential smoothing with trend and

regression analysis. The former is a type of double exponential smoothing thatallows for simultaneous smoothing on the series and on the trend. The latter is amethod that fits a straight line to a set of data.

4.3.1. Enhanced exponential smoothing with trend

Note that in simple exponential smoothing, the forecast always lags the actualoccurrence. This can be somewhat corrected by adding in a trend adjustment. Asmoothing constant delta ( δ ) is introduced.The equation to compute the forecast including trend (FIT) is

FITt = Ft + Tt

Where Ft = FITt-1 + α (At-1 – FITt-1), and Tt = Tt-1 + δ (Ft – FITt-1)

To get the equation going, the first time it is used the trend value must be enteredmanually. This initial trend value can be an educated guess or a computation basedon observed past data.

Example 4: Assume an initial starting forecast of 100, a trend of 10, an alpha of 0.2, and a deltaof 0.3. If actual demand turned out to be 115 rather than the forecast 100, calculatethe forecast including trend for the next period. If the actual for the next periodturned out to be 120, then what is the forecast including trend for the second nextperiod?


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4.3.2. Linear regression analysis

Regression can be defined as a functional relationship between two or morecorrelated variables. It is used to predict one variable given the other. Linearregression refers to the special class of regression where the relationship between

variables forms a straight line. The linear regression line if of the form Y = a + b x,where a is the Y intercept, b is the slope.

The least squares method is used to fit the line to the data. This method tries tominimize the sum of the squares of the vertical distance between each data pointand its corresponding point on the line. The parameters of the line are given by

∑∑

−

−=

−=

22

.

xn x

y xn xyb

xb ya

Example 5: A firm’s sales for a product line during the 12 quarters of the past three years wereas follows:

Quarter Sales Quarter Sales

1 600 7 2600

2 1550 8 2900

3 1500 9 3800

4 1500 10 4500

5 2400 11 4000

6 3100 12 4900

The firm wants to forecast each quarter of the fourth year—that is, quarters 13through 16.

Solution:x y xy x

2

1 600

2 1550

3 1500

4 1500

5 2400

6 3100

7 2600

8 2900

9 3800

10 4500

11 4000

12 4900


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4.4. Methods for seasonal series

A seasonal series is one that has a pattern that repeats every N periods for somevalue of N, which is referred to as the length of the season. Note that this is differentfrom the popular usage of the word season as a time of year. For example, the

demand for haircuts may peak on Saturday, week to week. In this case, theseasonal pattern lasts a week, and the seasons are the days of the week.

4.4.1. Seasonality

There are several ways to represent seasonality. Here we examine tow types ofseasonal variation: additive and multiplicative.

Additive seasonal variation simply assumes that the seasonal amount is aconstant no matter what the trend or average amount is. Seasonal forecastsare generated by adding a constant (say, 50 units) to the estimate of average

demand per season. In multiplicative seasonal variation, seasonal factors are multiplied by an

estimate of average demand to arrive at a seasonal forecast. This implies thatthe seasonal pattern depends on the level of demand. The peaks and valleysare more extreme when average demand is high. Essentially, this says thatthe larger the basic amount projected, the larger the variation around this thatwe can expect. We restrict our attention to this case as it is the usualexperience.

Seasonal factor (or index): a seasonal factor is the amount of correction needed in atime series to adjust for the season of the year.

4.4.2. Seasonal factors for stationary series

Now we present a simple method of computing seasonal factors for a time serieswith seasonal variation and no trend.

Compute the overall average per season from all the data Find the average demand for the same season Divide each seasonal average by the overall seasonal average. This gives

seasonal factors for each season.

To calculate each season’s forecast for next year, begin by estimating the averagedemand per season for next year, then obtain the final forecast by multiplying theseasonal factor by the average demand per season.

Example 6:The manager of the Stanley Steemer carpet cleaning company needs a quarterlyforecast of the number of customers expected next year. The carpet cleaningbusiness is seasonal, with a peak in the third quarter and a trough in the first quarter.Following are the quarterly demand data from the past four years:


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Quarter Year 1 Year 2 Year 3 Year 4

1 45 70 100 100

2 335 370 585 725

3 520 590 830 1160

4 100 170 285 215

The manager wants to forecast customer demand for each quarter of year 5, basedon her estimate of total year 5 demand of 2600 customers.

Solution:

Overall Avg quarterly sales in past years =

Avg quarterly sales for next year =

avg past salesseasonalfactor forecast for next year

Quarter 1

Quarter 2

Quarter 3

Quarter 4

4.4.3. Decomposition of a time series

When demand contains both seasonal and trend effects at the same time, we needto identify and separate the time series data into these components to obtain betterforecast. This is called decomposition of a time series.

Now let’s see how to decompose a time series using least squares regression. Thegeneral procedure involves 5 steps:

Step 1: determine the seasonal factor Step 2: deseasonalize the original data. To remove the seasonal effect on the

data, we divide the original data by the seasonal factor. Step 3: develop a least squares regression line for the deseasonalized data.

The purpose is to develop an equation for the trend line. Step 4: project the regression line through the period(s) to be forecasted. Step 5: create the final forecast by adjusting the regression line by the

seasonal factor (re-seasonalizing).

Example 7:


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Use the same data given in example 5, but now we consider the seasonal effect.Suppose the seasonal factor is given by the average for the same quarters in the 3-year period divided by the general average for all 12 quarters. What are theforecasts for the quarters 13 through 16?

Solution:

Overall quarterly avg. =

avg seasonal factor

Spring

Summer

Fall

Winter

Quarter Salesdeseasonalizeddemand

1 600

2 1,550

3 1,500

4 1,500

5 2,400

6 3,100

7 2,600

8 2,900

9 3,800

10 4,500

11 4,000

12 4,900

Quarter Trend-based

forecastFinal forecast

13

14

15

16


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4.5. Evaluating forecasts

4.5.1. Measures of forecast error

Define the forecast error in period t, e t, as the difference between the forecast value

for that period and the actual demand for that period:

et = Ft – At

Two common measures of forecast accuracy during n periods are the meanabsolute deviation (MAD) and the mean squared error (MSE), given by the followingformulas:

∑

∑

=

=

=

=

n

ii

n

i

i

en MSE

en

MAD

1

2

1

1

1

Note that the MSE is similar to the variance of a random sample. The MAD is oftenthe preferred method of measuring the forecast error because it does not requiresquaring. Furthermore, when forecast errors are normally distributed, as is generallyassumed, an estimate of the standard deviation of the forecast error, σe, is given by1.25 times the MAD.

Another measure of forecast accuracy is known as the mean absolute percentage

error (MAPE) and is given by 100*1

1

⎥⎦

⎤⎢⎣

⎡=

∑=

n

i i

i

A

e

n MAPE . It is independent of the

magnitude of the values of demand.

4.5.2. Criteria for selecting time-series methods

The criteria to use in making forecast method and parameter choices include Minimizing bias Minimizing MAD or MSE Meeting managerial expectations of changes in the components of demand Minimizing the forecast error last period.

However, managers recognize that the best technique in explaining the past data isnot necessarily the best to predict the future. For this reason, some analysts preferto use a holdout set as a final test. To do so, they set aside some of the more recentperiods from the time series, and use only the earlier time periods to develop andtest different model. Once the final models have been selected in the first phase,then they are tested again with the holdout set.


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5. Causal relationship forecasting

Linear regression technique is used in causal relationship forecasting. In the linearregression method, when the dependent variable (usually the vertical axis on agraph) changes as a result of time (plotted as the horizontal axis), it is time series

analysis. If one variable changes because of the change in another variable, this is acausal relationship (such as the number of deaths from lung cancer increasing withthe number of people who smoke).

Example 8:The Carpet City Store in Carpenteria has kept records of its sales (in square yards)each year, along with the number of permits for new houses in its area.

Year Permits Sales

1989 18 13000

1990 15 12000

1991 12 11000

1992 10 10000

1993 20 14000

1994 28 16000

1995 35 19000

1996 30 17000

1997 20 13000

Suppose that there are 25 permits for houses to be built in 2000. What is theforecast for sales in 2000?

6. Concluding remarks

In selecting a forecasting method to use, a firm should consider many factorsincluding time horizon to forecast, data availability, accuracy required, size of

forecasting budget, availability of qualified personnel, etc. Ways to cope with forecast errors: buffer—safety stock, safety lead time,

excess capacity. Characteristics of forecasts:

o They are usually wrong.o A good forecast is more than a single number.o Aggregate forecasts are more accurate.o The longer the forecast horizon, the less accurate the forecast will be.o Forecasts should not be used to the exclusion of known information.


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7. Exercises:

Question 1:

Sunrise Baking Company markets doughnuts through a chain of food stores. It has beenexperiencing over- and under-production because of forecasting errors. The following data are

its demand in dozens of doughnuts for the past four weeks. Doughnuts are made for thefollowing day; for example, Sunday’s doughnut production is for Monday’s sales, Monday’sproduction is for Tuesday’s sales, and so forth. The bakery is closed Saturday, so Friday’sproduction must satisfy demand for both Saturday and Sunday.

4 weeks ago 3 weeks ago 2 weeks ago last week

Monday 2200 2400 2300 2400

Tuesday 2000 2100 2200 2200

Wednesday 2300 2400 2300 2500

Thursday 1800 1900 1800 2000

Friday 1900 1800 2100 2000

SaturdaySunday 2800 2700 3000 2900

Make a forecast for this week on the following basis: A. Daily, using a simple four-week moving average.B. Daily, using a weighted average of 0.4, 0.3, 0.2, 0.1 for the past four weeks.C. Sunrise is also planning its purchases of ingredients for bread production. If bread demand

had been forecasted for last week at 22000 loaves and only 21000 loaves were actuallydemanded, what would Sunrise’s forecast be for this week using exponential smoothingwith alpha = 0.1?

D. Suppose, with the forecast made in c, this week’s demand actually turns out to be 22500.What would the new forecast be for the next week?

Question 2:

Here are quarterly data for the past two years. From these data, prepare a forecast for theupcoming year using decomposition.

Period Actual demand

1 300

2 540

3 8854 580

5 416

6 760

7 1191

8 760


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Question 3:

The demand for Krispee Crunchies, a favorite breakfast cereal of people born in the 1940s, isexperiencing a decline. The company wants to monitor demand for this product closely as itnears the end of its life cycle. The trend-adjusted exponential smoothing method is used withalpha = 0.1 and delta=0.2. At the end of December, the January estimate for the average

number of cases sold per month, FJan, was 900000 and the trend, TJan, was -50000 per month.The following table shows the actual sales history for Jan, Feb, and Mar. Generate forecast forFeb, Mar, and Apr.

Month Sales

Jan 890,000

Feb 800,000

Mar 825,000

Question 4:

The Northville Post Office experiences a seasonal pattern of daily mail volume every week. Thefollowing data for two representative weeks are expressed in thousands of pieces of mail:

Day week 1 week 2

Sunday 5 8

Monday 20 15

Tuesday 30 32

Wednesday 35 30

Thursday 49 45

Friday 70 70Saturday 15 10

total 224 210

A. Calculate a seasonal factor for each day of the week.B. If the postmaster estimates that there will be 230,000 pieces of mail to sort next week,

forecast the volume for each day of the week.


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Answer Key

Question 1:

A. Monday: (2200+2400+2300+2400)/4 = 2325 doz

Tue: 2125Wed: 2375Thu: 1875Fri: 1950Sat & Sun: 2850

B. Monday: (0.1*2200+0.2*2400+0.3*2300+0.4*2400)/4 = 2350 dozTue: 2160Wed: 2400Thu: 1900Fri: 1980Sat & Sun: 2880

C. Ft = Ft-1 + α (At-1 – Ft-1) = 22000+0.1*(21000-22000) = 21900D. Ft-1 = 21900 + 0.1*(22500-21900) = 21960

Question 2:

Period Actual demand period avg seasonal facto deseasonalized demand

1 300 358 0.527 568.99

2 540 650 0.957 564.09

3 885 1038 1.529 578.92

4 580 670 0.987 587.79

5 416 0.5272 789.01

6 760 0.9573 793.91

7 1191 1.5287 779.08

8 760 0.9867 770.21

Avg 679

Run a regression in Excel using deseasonalized demand, we obtain the parameter values: a = 500.6,b=39.64

Therefore, we have:

period trend forecast seasonal factor final forecast

9 857.4 0.527 452.0

10 897.0 0.957 858.7

11 936.6 1.529 1431.9

12 976.3 0.987 963.3


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Question 3:

Feb: 804800Mar: 755024

Apr: 714125

Question 4:

Day week 1week

2

daily

avgseasonal factor forecast

Sunday 5 8 6.5 0.210 6,889

Monday 20 15 17.5 0.565 18,548

Tuesday 30 32 31 1.000 32,857

Wednesday 35 30 32.5 1.048 34,447

Thursday 49 45 47 1.516 49,816

Friday 70 70 70 2.258 74,194Saturday 15 10 12.5 0.403 13,249

total 224 210 31


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8. Appendix


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Ch6 Forecasting

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