ECONOMICS FORECAST

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J

CHA PTER

MOVING AVERAGES AND SMOOTHING METHODS

This chapter will describe three simple approaches to forecasting a time series: naive, averaging, and smoothing methods. Naive methods are used to develop simple models that assume that very recent data provide the best predictors of the future. Averaging methods generate forecasts based on an average of past observations. Smoothing methods produce forecasts by averaging past values of a series with a decreasing ( exponential) series of weights.

Figure 4-1 outlines the forecasting methods discussed in this chapter. Visualize yourself on a time scale. You are at point tin Figure 4-1 and can look backward over past observations of the variable of interest (Yr) or forward into the future. After you select a forecasting technique, you adjust it to the known data and obtain forecast val- ues (Yi). Once these forecast values are available, you compare them to the known observations and calculate the forecast error (e,).

A good strategy for evaluating forecasting methods involves the following steps:

1. A forecasting method is selected based on the forecaster 's analysis of and intuition about the nature of the data.

2. The data set is divided into two sections-an initialization or fitting section and a test or forecasting section.

3. The selected forecasting technique is used to develop fitted values for the initial- ization part of the data.

4. The technique is used to forecast the test part of the data, and the forecasting error is determined and evaluated (refer to Chapter 3 for a review of measures of fore- casting accuracy).

5. A decision is made. The decision might be to use the technique in its present form, to modify the technique, or to develop a forecast using another technique and compare the results.

FIGURE 4-1 Forecasting Outline

where

Past data You are here

t

... Y,_3, Y,_2 , Y1_1, Y1 ,

Y, is the most recent observation of a variable A

Yr+i is the forecast for one period in the future

Periods to be forecast

A A A

Yr+l• Y1+2• Yt+3• · · ·

107

I'

108 CHAPTER 4 Moving Averages and Smoothing Methods

NAIVE MODELS

Often young businesses face the dilemma of trying to forec_ast with ~ery small ~ata sets. This situation creates a real problem, since many forecastmg techmques require large amounts of data. Naive forecasts are one possible solution, since they are based solely on the most recent information available.

Naive forecasts assume that recent periods are the best predictors of the future.

The simplest model is

½+1 = Yr (4.1)

where Yr+ 1

is the forecast made at time t (the forecast origin) for time t + 1. The naive forecast for each period is the immediately preceding observation. One

hundred percent of the weight is given to the current value of the series. The naive forecast is sometimes called the " no change" forecast. In short-term weather forecast- ing, the "no change" forecast occurs often. Tomorrow's weather will be much like today's weather.

Since the naive forecast (Equation 4.1) discards all other observations, this scheme tracks changes very rapidly. The problem with this approach is that random fluctua- tions are tracked as faithfully as other fundamental changes.

Example 4.1 Figure 4-2 shows the quarte rly sales of saws from 2000 to 2006 for the Acme Tool Company. The naive technique is used to forecast sales for the next quarter to be the same as for the previous quarter. Table 4-1 shows the data from 2000 to 2006. If the data from 2000 to 2005 are used as the initialization part and the data from 2006 as the test part, the forecast for the first quarter of 2006 is

Yz4+1 = Yz4 ¥25 = 650

FIGURE 4-2 Time Series Plot for Sales of Saws for Acme Tool Company, 2000- 2006, for Example 4 . 1

Sales of Saws for the Acme Tool Company: 2000--2006 900,----------------------,

BOO

700

600

500

400

300

200

100 ~~~ ~Q~l---Q~l:----Q,l _ __ Q•l---Qrl---Q~l---Q~l-_j Year 2000 2001 2002 2003 2004 2005 2006

CHAPTER 4 Moving Averages and Smoothing Methods 1 09

TABLE 4-1 Sales of Saws for Acme Tool Company, 2000-2006, for Example 4.1

Year Quarter Sa les

2000 1 1 500 2 2 350 3 3 250 4 4 400

2001 1 5 450 2 6 350 3 7 200 4 8 300

2002 1 9 350 2 10 200 3 11 150 4 12 400

2003 1 13 550 2 14 350 3 15 250 4 16 550

2004 1 17 550 2 18 400 3 19 350 4 20 600

2005 1 21 750 2 22 500 3 23 400 4 24 650

2006 1 25 850 2 26 600 3 27 450 4 28 700

The forecasting error is determined using Equation 3.6. The error for period 25 is

e25 = Yzs - Y25 = 850 - 650 = 200

In a similar fashion, the forecast for period 26 is 850, with an error of - 250. Figure 4-2 shows t?at these data have an upward ~rend ~nd that ther_e_ap~ears to be a seasonal pattern (the first and fourth quarters are relat1vely high), so a dec1s10n 1s made to modify the naive model.

Examination of the data in Example 4.1 leads us to conclude that the values are increasing over time. When data valu es increase over time, they are said to be nonstationary in level or to have a trend. If Equation 4.1 is used, the projections will be consistently low. However, the technique can be adjusted to take trend into considera- tion by adding the difference between this period and the last period. The forecast

equ ation is

¼+1 = Yr + (Yr - Yr-1) (4.2)

Equation 4.2 takes into account the amount of change that occurred between quarters.

11 0 CHAPTER 4 Moving Averages and Smoothing Methods

Example 4.1 (cont.) 200

6 · Using Equation 4.2, the forecast for the first quarter of is

Y24+1 = Y24 + (Y24 - Y24-1) Y 25 = Y24 + (Y24 - Y23) y25 = 650 + c 650 - 400) fi5 = 650 + 250 = 900

The forecast error with this model is

e25 = Yzs - Y = 850 - 900 = -50

For some purposes, the rate of change ~i~ht be more appropriate than the absolute amount of change. If this is the case, 1t 1s reasonable to generate forecasts

according to A Yr ½+1 = Yc--

Yr- 1 (4.3)

Visual inspection of the data in Table 4-1 indicates that seasonal varia~ion seems to exist. Sales in the first and fourth quarters are typically larger than those m the second and third quarters. If the seasonal pattern is strong, then an appropriate forecast equa- tion for quarterly data might be

½+1 = Yr-3 (4.4)

Equation 4.4 says that in next quarter the variable will take on the same value that it did in the corresponding quarter one year ago.

The major weakness of this approach is that it ignores everything that has occurred since last year and also any trend. There are several ways of introducing more recent information. For example, the analyst can combine seasonal and tre nd estimates and forecast the next quarter using

A (Y, - Yr-1) + • • • + (Y,-3 - Y,_4) Y, - Yi -4 (4.5) Yr +t = Yr- 3 +

4 = Y,-3 + 4

where the Yc-3 term forecasts the seasonal pattern and the remaining term averages the amount of change for the past four quarters and provides an estimate of the trend.

The naive forecasting models in Equations 4.4 and 4.5 are given for quarterly data. Adjustments can be made for data collected over different time periods. For monthly data, for example, the seasonal period is 12, not 4, and the forecast for the next period (month) given by Equation 4.4 is Yi+1 = Y1_11-

It is apparent that the number and complexity of possible naive models are limited only by the ingenuity of the a nalyst, but use of these techniques should be guided by sound judgment.

Naive methods are also used as the basis for making comparisons against which the performance of more sophisticated methods is judged.

Example 4. 1 (cont.) The forecasts for the first quarter of 2006 using Equations 4.3, 4.4, and 4.5 are

~ _ Y24 Y24 Y24+1 - Y24 - - = Y24 -

Y24- 1 Y23 A 650 Y2s = 650 - = 1 056

400 '

(Equation 4.3)

CHAPTER 4 Moving Averages and Smoothing Methods 1 11

Y24+1 = Yi4-3 = Yi1

Y25 = Y21 = 750 (Equation 4.4)

~ (Yz4 - Yz4-i) + ... + ( Yi4-3 - Yz4-4) Y24 - Yi-4 Yz4+1 = Y24_3 +

4 = Yz4-3 +

4 A ( Yz4 - Yio) 650 - 600

Yzs = Yz1 + ---- = 750 + - --- (Equation 4.5) 4 4

¥25 = 750 + 12.5 = 762.5

FORECASTING METHODS BASED ON AVERAGING

Frequently, management faces the situation in which forecasts need to be updated daily, weekly, or monthly for inventories containing hundreds or thousands of items. Often it is not possible to develop sophisticated forecasting techniques for each item. Instead, some quick, inexpensive, very simple short-term forecasting tools are needed to accomplish this task.

A manager facing such a situation is likely to use an averaging or smoothing tech- nique. These types of techniques use a form of weighted average of past observations to smooth short-term fluctuations. The assumption underlying these techniques is that the fluctuations in past values represent random departures from some underlying structure. Once this structure is identified, it can be projected into the future to pro- duce a forecast.

Simple Averages Historical data can be smoothed in many ways. The objective is to use past data to develop a forecasting model for future periods. In this section, the method of simple averages is considered. As with the naive methods, a decision is made to use the first t data points as the initialization part and the remaining data points as the test part. Next, Equation 4.6 is used to average (compute the mean of) the initialization part of the data and to forecast the next period.

A 1 I Yr+1 = - ~Y;

t i=l (4.6)

When a new observation becomes available, the forecast for the next period, Y,+2, is the average or the mean computed using Equation 4.6 and this new observation.

When forecasting a large number of series simultaneously ( e.g., for inventory man- agement), data storage may be an issue. Equation 4.7 solves this potential problem. Only the most recent forecast and the most recent observation need be stored as time moves forward.

A tY;+1 + y t+ l Y,+z = t + l (4.7)

The m ethod of simple averages is an appropriate technique when the forces gener- ating the series to be forecast have stabilized ~nd the envi~onment in whic~ the series exists is generally unchanging. Examples of this type of senes a~e the quantity of sales resulting from a consistent le~el of salesperson effort; the quantity ?f sales of a product in the mature stage of its life cycle; and the number of appomtments per week requested of a dentist, doctor, or lawyer whose patient or client base is fairly stable.

11 2 CHAPTER 4 Moving Averages and Smoothing Methods

• 1

the mean of all relevant historical observations as the A szmp e average uses forecast for the next period.

Example 4.2 . The Spokane Transit Authority operates a fleet of vans used to tran~port both pers?ns With disabilities and the elderly. A record of the gasoline purchased for th•~ fleet of ~ans 1s sh?wn in Table 4-2. The actual amount of gasoline cons1;1me? by a van '?n a _given day 1s det~rrruned by the random nature of the calls and the destmations. Exa~nation of the gasol~e pur- chases plotted in Figure 4-3 shows the data are very stable. Smee the data_ seem stationary, the method of simple averages is used for weeks 1 to 28 to forecast gasolme purchases for weeks 29 and 30. The forecast for week 29 is

TABLE 4-2 Gasoline Purchases for the Spokane Transit Authority for Example 4.2

Week t

1 2 3 4 5 6 7 8 9

10

310

300

290

~ C 0 'is

280

c:, 270

260

250

240

Gallons Week Y, t

275 11 291 12 307 13 281 14 295 15 268 16 252 17 279 18 264 19 288 20

Gallons Y,

302 287 290 311 277 245 282 277 298 303

Week t

21 22 23 24 25 26 27 28 29 30

Gallons Y,

310 299 285 250 260 245 271 282 302 285

Gasoline Purchases for Spokane Transit Authority

3 6 9 12 15 18 21 24 27 Week

FIGURE 4 -3 Time Series Plot of Weekly Gasoline Purchases for the Spokane Transit Authority for Example 4 .2

30

CHAPTER 4 Moving Averages and Smoothing Methods 113

The forecast error is

A 1 28 Yzs+1 = -LY;

28 ;=] A 7,874 Y29 = 28 = 281.2

e29 = Yz9 - Y29 = 302 - 281.2 = 20.8

. The forecast for week 30 includes one more data point (302) added to the initialization penod. The forecast using Equation 4.7 is

f: _ 28¥28+1 + Y 2s + 1 28Y29 + Y29 28 + 2 - 28 + 1 29

y _ 28(281.2) + 302 _ 30 - 29 - 281.9

The forecast error is

e30 = Y30 - Y:30 = 285 - 281.9 = 3.1

Using the method of simple averages, the forecast of gallons of gasoline purchased for week 31 is

A 1 30 8,461 Y:io+1 = - LY; = -- = 282

30i=l 30

Moving Averages The method of simple averages uses the mean of all the data to forecast. What if the analyst is more concerned with recent observations? A constant number of data points can be specified at the outset and a mean computed for the most recent observations. The term moving average is used to describe this approach. As each new observation becomes available, a new mean is computed by adding the newest value and dropping the oldest. This moving average is then used to forecast the next period. Equation 4.8 gives the simple moving average forecast. A moving average of order k, MA(k), is computed by

A }; + Yi-1 + . + Yi'- k+l Yi+ 1 = --'----'--=---k---_;__~

where

Yi+ 1 = the forecast value for the next period Yi = the actual value at period t k = the number of terms in the moving average

(4.8)

The moving average for time period t is the arithmetic mean of the k most recent observations. In a moving average, equal weights are assigned to each observation. Each new data point is included in the average as it becomes available, and the earliest data point is discarded. The rate of response to changes in the underlying data pattern depends on the number of periods, k, included in the moving average.

Note that the moving average technique deals only with the latest k periods of known data; the number of data points in each average does not change as time advances. The moving average model does not handle trend or seasonality very well, although it does better than the simple average method.

11 4 CHAPTER 4 Moving Averages and Smoothing Methods

h the number of periods k, in a moving average. A moving

Toe analyst must c oose ' . d · f f d 1 MA(l) would take the current observation, Y,, an use 1t to orecast

average o or er .' d Thi '·s sllll· ply the naive forecasting approach of Equation 4.1. Y for the next peno . s 1

· f order k 1·s the mean value of k consecutive observations. The A moving average o . most recent moving average value provides a forecast for the next penod.

Example 4.3 . . · h h S k 11 · Table 4-3 demonstrates the moving average forecastrng te~hnique wit t e po ane rans~t Authority data, using a five-week moving average. The movrng average forecast for week 29 IS

A y28 + Y28- l + . 0 0 + Y28-5+1 Yzs+ l = 5

TABLE 4-3 Gasoline Purchases for the Spokane Transit Authority for Example 4 .3

t Gallons Y, e,

1 275 2 291 3 307 4 281 5 295 6 268 289.8 -21.8 7 252 288.4 -36.4 8 279 280.6 -1.6 9 264 275.0 - 11.0

10 288 271.6 16.4 11 302 270.2 31.8 12 287 277.0 10.0 13 290 284.0 6.0 14 311 286.2 24.8 15 277 295.6 -18.6 16 245 293.4 - 48.4 17 282 282.0 0.0 18 277 281.0 - 4.0 19 298 278.4 19.6 20 303 275.8 27.2 21 310 281.0 29.0 22 299 294.0 5.0 23 285 297.4 - 12.4 24 250 299.0 -49.0 25 260 289.4 - 29.4 26 245 280.8 -35.8 27 271 267.8 3.2 28 282 262.2 19.8 29 302 261.6 40.4 30 285 272.0 13.0

CHAPTER 4 Moving Averages and Smoothing Methods 1 1 5

y,; _ 282 + 271 + 245 + 260 + 250 1,308 29 - = -- = 261 6 5 5 .

When the actual value for week 29 is known, the forecast error is calculated:

e29 = Yz9 - }z9 = 302 - 261.6 = 40.4

The forecast for week 31 is

A Y30 + Y:io-1 + · · · + Y:io-s+1 Y:io+1 = _______ ___:c.:.......::........:. 5

y.; _ Y30 + Yz9 + Yzs + Yi1 + Yi6 31 - 5

151 = 285 + 302 + 282 + 271 + 245 = 1,385 = 277 5 5

~in~tab can_ be used to compute a five-week moving average (see the Minitab Apphcat1ons section at the end of the chapter for instructions). Figure 4-4 shows the five- week moving average plotted against the actual data; the MAPE, MAD, and MSD; and the basic Minitab instructions. Note that Minitab calls the mean squared error MSD (mean squared deviation).

Figure 4-5 shows the autocorrelation function for the residuals from the five-week moving average method. Error limits for the individual autocorrelations centered at zero and the Ljung-Box Q statistic (with six degrees of freedom, since no model parameters are estimated) indicate that significant residual autocorrelation exists. That is, the residuals are not random. The association contained in the residuals at certain time lags can be used to improve the forecasting model.

Toe analyst must use judgment when determining how many days, weeks, months, or quarters on which to base the moving average. The smaller the number, the larger the weight given to recent periods. Conversely, the larger the number, the smaller the weight given to more recent periods. A small number is most desirable when there are

FIGURE 4-4 Five-Week Moving Average Applied to Weekly Gasoline Purchases for the Spokane Transit Authority for Example 4.3

310

300

290

~ 280 0

'lti c:, 270

260

250

240 3 6

Moving Average Plot for Gallons

~ , ' I ""

9 12 15 Weeks

18 21 24 'Z7 30 Mi1itab Jnstru:tions

__ .. , I MoYng Avenge

Length S

Ac.cur «y MHSU'tS MAPE 7,503

MAO 20,584

MSO 622,14'l

st.at> Time Series>Moyro Averages

116 CHAPTER 4 Moving A verages and Smoothing Methods

Autocorrelation Function: MA(5) Residuals

Lag ACF T LBQ

l 0 .506 290 2 . 5 3 7.21

2 o. 078 551 0 . 3 2 7. 39 3 -0. 298611 - 1. 21 10 . 13

4 -0 . 60 2830 -2 . 31 21. 81

5 -0. 64263 2 - 2 .06 35 . 74

6 -0. 219452 - 0 .61 37.46

1.0

0.8

O.G

C: 0.4

·i 0.2

Autocorre:l~tion Function for MA(5) Residu.tls

____ __ .,. .--___ , __ __ ..

l o.o.J.L--.-L---.---.---,--.1 ~ .().2

I -0.4 -0.G

-0 .8

- --------- --- -1.0 L----2.-------,,----◄r----;r---,-'

l•9

FIGURE 4-5 Autocorrelation Function for the Residuals When a Five-We~k Moving Average Method Was Used with the Spokane Transit Authority Data for Example 4.3

sudden shifts in the level of the series. A small number places heavy weight on recent history, which enables the forecasts to catch up more rapidly to t_he c~rrent le~el. A large number is desirable when there are wide, infrequent fluctuations 10 the sene&

Moving averages are frequently used with quarterly or monthly data to help smooth the components within a time series, as shown in Chapter 5. For quarterly data, a four- quarter moving average, MA( 4 ), yields an average of the four quarters, and for monthly data, a 12-month moving average, MA(12) , e liminates o r averages o ut the seasonal effects. The larger the order of the moving average, the greater the smoothing effect.

In Example 4.3, the moving average technique was used with stationary data._In Example 4.4, we show what happens when the moving average method is used with trending d ata. The double moving average technique, which is designed to handle trending data, is introduced next.

Double Moving Averages One way of forecasting time series data that h ave a linear trend is to use double mo~- ing averages. This method does what the name implies: One set of moving averages 15

computed, and the n a second set is compute d as a moving average of the first set. Table 4-4 shows the weekly rentals data for the Movie Video Store along with the

results of using a three-week moving average to fo recast future sales. Examination of the error column in Table 4-4 shows that every entry is positive, signifying that the forecasts do not catch up to the trend. The three-week moving average and the double moving average for these data are shown in Figure 4-6. Note h ow the three-week rool'· ing averages lag behind the actual values for comparable periods. This illustrates what happens when the moving average technique is used with trending data. Note also that the double moving averages lag behind the first set about as much as the first set lags behind the actual values. The difference between the two sets of moving averages is added to the three-week moving average to forecast the actual values.

Equations 4.9 through 4.12 summarize double moving average construction. First. Equation 4.8 is used to compute the moving average of order k.

M _ Y.A _ }; + ¼ - 1 + ¼ -2 + . . . + }; k+ l I - t+ l - k

u,

~ C Q)

a:

CHAPTER 4 Moving Averages and Smoothing Methods 1 1 7

TABLE 4-4 Weekly Rentals for the Movie Video Store for Example 4.4

Weekly Units Three-Week Moving Av~rage t Rented Y, Moving Total Forecast Y,+1 e

1 654 2 658 3 665 1,977 4 672 1,995 659 13 5 673 2,010 665 8 6 671 2,016 670 1 7 693 2,037 672 21 8 694 2,058 679 15 9 701 2,088 686 15

10 703 2,098 696 7 11 702 2,106 699 3 12 710 2,115 702 8 13 712 2,124 705 7 14 711 2,133 708 3 15 728 2,151 711 17 16 717

MSE = 133

730

720

710 A" :x· ,.........,:.~-· -~ 700 .... - ;x··

#' - •• / •• x

690 / . -I .><

/ . 680 /,;;,,/

~.x·· 670 ,,,.- .><

. ;Ir.,,. :.t• ' .,,.

660 -t"'

650

5 10 15 Week

Rentals

Moving Average

Double Moving Average

FIGURE 4-6 Three-Week Single and Double Moving Averages for the Movie Video Store Data for Example 4.4

The n E qua tion 4.9 is used to compute the second moving average.

M, + M,-1 + M,-2 + ... + M,- k+ l M; = k (4.9)

Equation 4.10 is used to develop a forecast by add~g to the single moving average the difference b e tween the single and the second movmg averages.

a 1 = Mi + (M, - M ; ) = 2M1 - M ; (4.10)

118 CHAPTER4 Moving Averages and Smoothing Methods

Equation 4.11 is an additional adjustment factor , which is similar to a slope measure

that can change over the series.

b, = k : l (M, - M/) (4.11)

Finally, Equation 4.12 is used to make the forecast p periods into the future.

½+p =a,+ b,p

where k = the number of periods in the moving average

p = the number of periods ahead to be forecast

(4.12)

Example 4.4 The Movie Video Store operates several videotape rental outlets in Denver, Colorado.The company is growing and needs to expand its inventory to accommodate the increasing demand for its services. The president of the company assigns Jill Ottenbreit to forecast rentals for the next month. Rental data for the last 15 weeks are available and are presented in Table 4-5. At first, Jill attempts to develop a forecast using a three-week moving average. The MSE for this model is 133 (see Table 4-4). Because the data are obviously trending,she finds that her forecasts are consistently underestimating actual rentals. For this reason,she decides to try a double moving average. The results are presented in Table 4-5. To under- stand the forecast for week 16, the computations are presented next. Equation 4.8 is used to compute the three-week moving average (column 3).

A Yi s + Yis-1 + Yis-3+1 M1s = Yis +1 =

3

728 + 711 + 712 ___ 3 ___ = 717

TABLE 4-5 Double Moving Average Forecast for the Movie Video Store for Example 4.4

(1) (2) (3) Three-Week (4) Double (5) (6) (7) Time Weekly Moving Moving Average Value Value Forecast

t Sales Y, Average M, M ' I ofa ofb a+ bp (p = 1)

1 654 2 658 3 665 659 4 672 665 5 673 670 665 675 5 6 671 672 669 675 3 680 7 693 679 674 684 5 678 8 694 686 679 693 7 689 9 701 696 687 705 9 700

10 703 699 694 704 5 . 714 11 702 702 699 705 3 709 12 710 705 702 708 3 708 13 712 708 705 711 3 711 14 711 711 708 714 714 3 15 728 717 712 722 717 16

5 727

MSE = 63.7

(8) e,

- 9 15 5 1

-11 -7 2 1

-3 11

CHAPTER 4 Moving Averages and Smoothing Methods

Then Equation 4.9 is used to compute the do uble moving average (column 4).

M , _ M1s + M1s- 1 + M1s-3+1 15 - 3

M ' 717 + 711 + 708 15 = 3 = 712

119

Equation 4.10 is used to compute the difference between the two moving averages (column 5).

a1s = 2M15 - M;5 = 2(717) - 712 = 722 Equation 4.11 is used to adjust the slope (column 6).

Equation 4.12 is used to make the forecast one period into the future (column 7).

Yis+1 = a1s + b1sP = 722 + 5(1) = 727

The forecast four weeks into the future is

Yis+4 = a 1s + b1sP = 722 + 5(4) = 742

Note that the MSE has been reduced from 133 to 63.7.

It seems reasonable that more-recent observations are likely to contain more- importa nt information. A procedure is introduced in the next section that gives more e mphasis to the most recent observations.

EXPONENTIAL SMOOTHING METHODS

Whereas the m ethod of moving averages takes into account only the most recent observations, simple exponential smoothing provides an exponentially weighted mov- ing average of all previously observed values. The model is often appropriate for data with n o predictable upward or downward trend. The aim is to estimate the current leve l. This level estimate is then used as the forecast of future values.

Exponential smoothing continually revises an estimate in the light of more-recent experiences. This method is based on averaging (smoothing) past values of a series in an exp onentially decreasing manner. The most recent observation receives the largest weight, a (where O < a < l); the next most recent observation receives less weight, a(l - a); the observation two time periods in the past receives even less weight, a(l - a)2; and so forth.

In one representation of exponential smoothing, the new forecast (for time t + 1) may be thought of as a weighted sum of the new observation (at time t) and the old forecast (for time t). The weight a (0 < a < 1) is given to the newly observed value, and the weight (1 - a) is given to the old forecast. Thus,

New forecast= [a X (New observation) ] + [(1 - a) X (Old forecast)]

More formally, the exponen tia l smoothing equation is

Yi'+1 = aY, + (1 - a)Y, (4.13)

~

1 2 0 CHAPTER 4 Moving Averages and Smoothing Methods

where Yi+1 = the new smoothed value or the forecast value for the next period

a = the smoothing constant (0 < a < 1) Y, = the new observation or the actual value of the series in period t

I

Y, = the old smoothed value or the forecast for period t

Equation 4.13 can be written as

½+1 = aYc + (1-a)"Y, = aYc + "Y, - a"Y, ½+1 = "Y, + a(Yc - Yi)

In this form, the new forecast CY,+ 1) is the old forecast (Y1) adjusted by a times the error Y, - Y, in the old forecast.

In Equation 4.13, the smoothing constant, a, serves as the weighting factor. The value of a determines the extent to which the current observation influences the fore- cast of the next observation. When a is close to 1, the new forecast will be essentially the current observation. (Equivalently, the new forecast will be the old forecast plus a substantial adjustment for any error that occurred in the preceding forecast.) Conversely, when a is close to zero, the new forecast will be very similar to the old fore- cast, and the current observation will have very little impact.

Exponential smoothing is a procedure for continually revising a forecast in the light of more-recent experience.

Finally, Eq~ation 4.13 implies, for time t, that Yr = Clfc_1 + (1 - cx)Yr-1, and substi· tution for Yc in Equation 4.13 gives

Yi+1 = aYc + (1 - a)"Y, = aYc + (1 - a)[aYc- i + (1 - a)Y,-d ½+1 = aYc + a(l - a)Yc-1 + (1 - a)2"f;'_1

Continued subst~tution (for ¥1_ 1 and so forth) shows ¥1+ 1 can be written as a sum of current and previous Y's with exponentially declining weights or

½+1 = aYc + a(l - a)Yc'-i + a(l - a)2Yc_2 + a(l - a)3Yc_3 +... (4.14)

That is, Yr+l is an exponentially smoothed value. Tue speed at which past observations lose their impact depends on the value of a, as demonstrated in Table 4-6.

Equations 4.13 !ind 4.14 are equivalent, but Equation 4.13 is typically used to cal· culate the forecas! Y1+1 beca~se it requires less data storage and is easily implemented.

The value assigned to a is the key to the analysis. If it is desired that predictions be stable and random variations be smoothed, a small value of a is required. If a rapid res pons~ to a real change in the patt~rn of observations is desired, a larger value of a is appropnate. One method of ~strmatmg a is an iterative procedure that minimizes the mean squared error (MSE) given by Equation 3.8. Forecasts are computed for, say, a equal to .1, .2, ... , .9, and the sum of the squared forecast errors is computed for each.

CHAPTER 4 Moving Averages and Smoothing Methods 1 21

TABLE 4-6 Comparison of Smoothing Constants

a= .1 a= .6 Period Calculations Weight Calculations Weight

t .100 .600 t-1 .9 X .1 .090 .4 X .6 .240 t-2 .9 X .9 X .1 .081 .4 X .4 X .6 .096 t- 3 .9 X .9 X .9 X .1 .073 .4 X .4 X .4 X .6 .038 t-4 .9 X .9 X .9 X .9 X .1 .066 .4 X .4 X .4 X .4 X .6 .015 All others .590 .011

Totals 1.000 1.000

The value of a producing the smallest error is chosen for use in generating future forecasts.

To start the algorithm for Equation 4.13, an initial value for the old smoothed series must be set. One approach is to set the first smoothed value equal to the first observation. Example 4.5 will illustrate this approach. Another method is to use the average of the first five or six observations for the initial smoothed value.

Example 4.5 The exponential smoothing technique is demonstrated in Table 4-7 and Figure 4-7 for Acme Tool Company for the years 2000 to 2006, using smoothing constants of .1 and .6. The data for the first quarter of 2006 will be used as test data to help determine the best value of a (amop.g the two considered). The exponentially smoothed series is computed by initially set- ting Yi equal to 500. If earlier data are available, it might be possible to use them to develop a smoothed series up to 2000 and then use this experience as the initial value for the smoothed series. The computations leading to the forecast for periods 3 and 4 are demon- strated next.

1. Using Equation 4.13, at time period 2, the forecast for period 3 with a. = .1 is

Yz+1 = aYi + (1 - a)Yz Yj = .1(350) + .9(500) = 485 .

2. The error in this forecast is

e3 = ~ - Yj = 250 - 485 = - 235

3. The forecast for period 4 is

Z+1 = a~ + (1 - a)Yj ¼ = .1 (250) + .9(485 ) = 461.5

From Table 4 -7, when the smoothing constant is .1, the forecast for the first quarte r of 2006 is 469, with a squared error of 14?,161. When the smoothing constant is .6, the forecast for the first quarter of 2006 is 576, with a squared error of 75,076. On the basis of this limited evidence, exponential smoothing with a = .6 performs better than exponential smoothing with a= .1.

In Figure 4-7, note how stable the smoothed values are for the .1 smoothing con- stant. On the basis of minimizing the mean squared error, MSE (MSE is called MSD on the Minitab output), over the first 24 quarters, the .6 smoothing constant is better.

1 2 2 CHAPTER 4 Moving Averages and Smoothing Methods

TABLE 4-7 Exponentially Smoothed Values for Acme Tool Company Sales for Example 4 . 5

Actual Time Value

Year Quarters Y,

Smoothed Forecast

Value Error

:i\(a = .1) e,

Smoothed Value

Y/a = .6)

2000 1 500

2 350

500.0 0.0

500.0 -150.0

500.0 500.0

3 250 485.0 (1) -235.0 (2) 410.0

4 400 461.5 (3) -61.5 314.0

2001 5 450 455.4 -5.4 365.6

6 350 454.8 - 104.8 416.2

7 200 444.3 -244.3 376.5

8 300 419.9 - 119.9 270.6

2002 9 350 407.9 -57.9 288.2

10 200 402.1 -202.1 325.3

11 150 381.9 -231.9 250.1

12 400 358.7 41.3 190.0

2003 13 550 362.8 187.2 316.0

14 350 381.6 -31.5 456.4

15 250 378.4 - 128.4 392.6

16 550 365.6 184.4 307.0

2004 17 550 384.0 166.0 452.8

18 400 400.6 - 0.6 511.1

19 350 400.5 -50.5 444.5

20 600 395.5 204.5 387.8

2005 21 750 415.9 334.1 515.1

22 500 449.3 -50.7 656.0 23 400 454.4 -54.4 562.4 24 650 449.0 201.0 465.0

2006 25 850 469.0 381.0 576.0

Forecast Error

e,

0.0 -150.0

-160.0

86.0 84.4

-66.2

-176.5

29.4

61.8

-125.3

-100.1

210.0 234.0

-106.4

-142.6

243.0

97.2

-111.l

-94.5

212.2

234.9

-156.0

-162.4

185.0

274.0

Noce:The numbers in parentheses refer to the explanations given in the text in Example 45.

If the m~an absolute percentage errors (MAP Es) are compared, the .6 smoothing constant IS also better. To summarize:

a= .l

a = .6

MSE = 24,262 MSE = 22,248

MAPE = 38.9% MAPE = 36.5 %

Ho"."e_ver: t~e MSE and MAPE are both large, and on the basis of these summary statistics, It IS apparent tha~ exponential smoothing does not represent these data well. As w_e ~hall see, a smoothmg method that allows for seasonality does a better job of predictmg the Acme Tool Company saw sales. _ A fac~or, other ~h~~ the chois_e of a, that affects the values of subsequent forecasts 1s the ~h~Ice of the lllltlal valu_e,_~1 for the smoothed series. In Table 4-7 (see Example 4.5) , Yi -: J-; "."as used as the m1tial smoothed value. This choice tends to give Y1 too ~uch weight m l~ter forecasts. Fortunately, the influence of the initial forecast dunin· 1shes greatly as t mcreases.

; (U

CHAPTER 4 Moving Averages and Smoothing Methods 1 23

Single Exponential Smoothing Plot for Saws soo-r------------------

Min itab Instructions 700 Stat> Time Series>Single Exp. smoothing

600

500

Variable --Actual ~Fits

Smoolhing Constant Alpha 0.1

Accuracy Measures MAPE 38.9

MAD 127.0 MSD 24261.7

en 400

300

200

100.._...,.-,----,--.--.--,----.----,~~-.-~-....J 2 4 6 8 10 12 14 16 18 20 22 24

Quarter

Single Exponential Smoothing Plot for Saws

800 V•ri•ble -Actu•I

700 ---- Fits Smoothing Constant

600 Alph• o.&

Accuracy Measures MAPE 3o.S

Cl) 500 :it

MAD 134,5 MSD 22248.4

(0 er.,

400

300

200

100 2 4 6 8 10 12 14 16 18 20 22 24

Quarter

FIGURE 4 -7 Exponential Smoothing for Acme Tool Company Data from Example 4.5: (Top) er = .1 and (Bottom) a = .6

Another approach to initializing the smoothing procedure is to average the first k observations. The smoothing then begins with

~ 1 k Yi= -k ~Y,

t=I

Often k is chosen to be a relatively small number. For example, the default approach in Minitab is to set k = 6.

1 2 4 CHAPTER 4 Moving Averages and Smoothing Methods

Example 4.6 The computation of the initial value as an average for the Acme Too! <;?mpany ~ata presented in Example 4.5 is shown next. If k is chosen to equal 6, then the 1mt1al value IS

Yi = l_ ±Yr= l.(500 + 350 + 250 + 400 + 450 + 350) = 383.3 6 1=1 6 .

The MSE and MAPE for each a when an initial smoothed value of 383.3 is used are shown next.

a= .1

a= .6

MSE = 21,091 MSE = 22,152

MAPE = 32.1% MAPE = 36.7%

The initial value, Yi = 383.3, led to a decrease in the MSE and MAPE for a = .1 but did not have much effect when a = .6. Now the best model, based on the MSE and MAPE sum- mary measures, appears to be one that uses a = .1 instead of .6.

Figure 4-8 shows results for Example 4.5 when the data are run on Minitab (see the Minitab Applications section at the end of the chapter for instructions). The smoothing con- stant of a = .266 was automatically selected by minimizing the MSE. The MSE is reduced to 19,447, the MAPE equals 32.2% , and although not shown, the MPE equals -6.4%. The forecast for the first quarter of 2006 is 534.

Figure 4-9 shows the autocorrelation function for the residuals of the exponential smoothing method using an alpha of .266. When the Ljung-Box test is conducted for six time lags, the large value of LBQ (33.86) shows that the first six residual autocorrelations as a group are larger than would be expected if the residuals were random. In particular, the significantly large residual autocorrelations at lags 2 and 4 indicate that the seasonal varia- tion in the data is not accounted for by simple exponential smoothing.

Exponential smoothing is often a good forecasting procedure when a nonrandom time series exhibits trending behavior. It is useful to develop a measure that can be used to determine when the basic pattern of a time series has changed. A tracking sig- nal is one way to monitor change. A tracking signal involves computing a measure of the forecast errors over time and setting limits so that, when the errors go outside those limits, the forecaster is alerted.

800

700

600

"' 500

3: ~ co 400

300

200

100

Single Exponential Smoothing Plot for Saws

2 4 6 8 ill ~ M ~ IB ~ ~ ~ Quarter

1~~:- I I Smoolhr,g Constant I

Alph, 0-266357

AccLr acy Mean.res MAPE 32.2

MAO 117.S

M5D 1'1+17,0

FIGURE 4-8 Exponential Smoothing with n = .266 for Acme Tool Company Data for Example 4.6

CHAPTER 4 Moving Averages and Smoothing Methods 1 25

..:. AutocorrelatlOO fot Residuals .~ .. ':""~..,,

Autocorrelation Function: Residuals Autocorrelation Function for Example 4 .6 Residuals

Laq ACF T l 0 . 121421 0.59 2 -0 . 588941 -2.84 3 0.109122 0.41 4 0.646967 2.40 5 -o. 006908 -0.02 6 -0.542625 -l.65

LBQ 0 . 40

10.24 10 . 59 23 . 65 23.65 33.86

I.D

0.8

0.6

-0.6

•0,8

·J.0

_____ .. --------- --------- I

,._ __ --------------------

I

FIGURE 4-9 Autocorrelation Function for the Residuals When Exponential Smoothing with o = .266 Is Used with Acme Tool Company Data for Example 4.6

A tracking signal involves computing a measure of forecast errors over time and setting limits so that, when the cumulative error goes outside those limits, the forecaster is alerted.

For example, a tracking signal might be used to determine when the size of the smoothing constant a should be changed. Since a large number of items are usually being forecast, common practice is to continue with the same value of a for many peri- ods before attempting to determine if a revision is necessary. Unfortunately, the sim- plicity of using an established exponential smoothing model is a strong motivator for not making a change. But at some point, it may be necessary to update a or abandon exponential smoothing altogether. When the model produces forecasts containing a great deal of error, a change is appropriate.

A tracking system is a method for monitoring the need for change. Such a system contains a range of permissible deviations of the forecast from actual values. As Jong as forecasts generated by exponential smoothing fall within this range, no change in a is necessary. However, if a forecast falls outside the range, the system signals a need to update a.

For instance, if things are going well, the forecasting technique should over- and underestimate equally often. A tracking signal based on this rationale can be developed.

Let U equal the number of underestimates out of the last n forecasts. In other words, U is the number of errors out of the last k that are positive. If the process is in control, the expected value of U is k /2, but sampling variability is involved, so values close to k /2 would not be unusual. On the other hand, values that are not close to k/2 would indicate that the technique is producing biased forecasts.

Example 4.7 Suppose that Acme Tool Company has decided to use the exponential smoothing technique with a equal to .1, as shown in Example 4.5 (seep. 121). If the process is in control and the analyst decides to monitor the last 10 error values, U has an expected value of 5. Actually,

1 26 CHAPTER 4 Moving Averages and Smoothing Methods

a u value of 2, 3, 4, 6, 7, or 8 would not be unduly ~la_rming. However, a value of 0, 1, 9, or 10 would be of concern, since the probability of obtamm~ such a ".alue by ch~nce alone would be .022 (based on the binomial distribution) . With this mformation, a trackmg system can be developed based on the following rules:

If 2 s U s 8, then the process is in control.

If U < 2 or U > 8 then the process is out of control.

Assume that out of the next 10 forecasts using this technique, only one has a positive error. Since the pr~bability of obtaining only one positive error out of 10 is quite low (.01), the process is considered to be out of control ( overestimating), and the value of a should be changed.

Another way of tracking a forecasting technique is to determine a range that should contain the forecasting errors. This can be accomplished by using the MSE that was established when the optimally sized a was determine d . If the exponential smoothing technique is reasonably accurate, the fore cast error should be approxi- mately normally distributed about a mean of ze ro. Under this condition, there is about a 95% chance that the actual observation will fall within approximately two standard deviations of the forecast. Using the RMSE as an estimate of the standard deviation of the forecast error, approximate 95% error limits can be determined. Forecast errors falling within these limits indicate no cause for alarm. Errors (particularly a sequence of errors) outside the limits suggest a change. Example 4.8 illus trate s this approach.

Example 4.8 In Example 4.6, on the Acme Tool Company, the optimal a was determined to be a = .266, with MSE = 19,447. An estimate of the standard de viation of the forecast errors is RMSE = "V19,M7 = 139.5. If the forecast errors are approximately normally distributed about a mean of zero, there is about a 95% chance that the actual observation will fall within two standard deviations of the forecast or within

±2RMSE = ±2\/'19,477 = ±2(139.5 ) = ± 279

For this example, the permissible absolute error is 279. If for any future forecast the magni- tude of the error is greater than 279, there is reason to believe that the optimal smoothing constant a should be updated or a different forecasting method considered.

The preceding discussion on tracking signals also applies to the smoothing meth- ods yet to be discussed in the rest of the chapter.

Simple exponential smoothing works well when the data vary about an infre- quently_ changing level. Whenever a sustained trend exists, exponential smoothing will lag_be~md t~e actual values over time. Holt's linear exponential smoothing technique, which is designed to handle data with a well-defined trend address es this problem and is introduced next. '

Exponential Smoothing Adjusted for Trend: Holt's Method In s~ple exponential s~oothing, the level of the time series is assumed to be changing occas1onally, an~ an estimate of t~e current level is required. In some situations, the ~bserved data will be clearly trendmg and contain information that allows the anticipa- t10n of future upward movements. When this is the case a linear trend forecast function is needed.~~~ business ~nd economic series rarely exhibit a fixed linear trend, we consider the poss1bilit~ of mode1:ing evolving local linear trends over time. Holt (2004) developed an ex~onential_ smoothing method, Holt's linear exponential smoothing 1 that allows for evolvmg local linear trends in a time series and can be used to generate forecasts.

1Holt's linear exponential smoothing is sometimes called double e · l h . xponentta sm oot mg.

CHAPTER 4 Moving A verages and Smoothing Methods 1 2 7

When a trend in the tim · · · · · e senes 1s anticipated, an estrmate of the current slope as ~ell as the cu_rrent level, is required. Holt 's technique smoothes the level and sl~pe dire~tly by _usmg different smoothing constants for each. These smoothing constants pro~ide estrmates of level and slope that adapt over time as new observations become ava~a_b_le. ?ne of t?e advantages of Holt's technique is that it provides a great deal of flexibility m selectmg the rates at which the level and trend are tracked.

The three equations u sed in Holt's method are

1. The exponentially smoothe d series, or current level estimate

L r = a}'r + (l - u)( L1-i + 7;_i) 2. The trend estimate

Tc = B(L1 - L1- 1) + (1 - ~)Ti-i 3. The forecast for p periods into the future

½+p = Lr + P'I'r where

L 1 = the new smoothed value ( e stimate of current level)

a = l the smoothing constant for the level (0 < a < 1) ½ = the new observation or actual value of the series in period t {3 = the smoothing constant for the trend estimate (0 < {3 < l )

7; = the trend estimate p = the periods to be forecast into the future

Yi+p = the forecast for p periods into the future

(4.15)

(4.16)

(4.17)

Equation 4.15 is very similar to the equation for simple exponential smoothing, Equation 4.13, except that a term ('I'r- 1 ) has been incorporated to properly update the level when a trend exists. That is, the current level (L 1 ) is calculated by taking a weighted aver- age of two estimates of level - one estimate is given by the current observation (Yi), and the other estimate is given by adding the previous trend CY;-1) to the previously smoothed level (L

1 _

1 ). If there is no trend in the data, there is no need for the term 7;_1 in Equation

4.15, effectively reducing it to Equation 4.13. There is also no need for Equation 4.16. A second smoothing constant, {3, is used to create the trend estimate. Equation 4.16

shows that the current trend (Tr) is a weighted average (with weights f3 and 1 - /3) of two trend estimates-one estimate is given by the change in level from time t - l to t ( L

1 - L r- i ) , and the other estimate is the previously smoothed trend (7;_1 ) . Equation

4.16 is similar to Equation 4.15, except that the smoothing is done for the trend rather

than the actual data. Equation 4.17 shows the forecast for p periods into the future. For a forecast made

at time t, the current trend estimate (7;) is multiplied by the number of periods to b e forecast ( p) , and then the product is added to the current level ( L 1) . Note that the fore- casts for future periods lie along a straight line with slope 7; and intercept L 1•

As with simple exponential smoothing, the smoothing constants a and f3 can be selected subjectively or generated by minimizing a measure of forecast error such as the MSE. Large weights r e sult in more rapid changes in the component; small weights result in less rapid changes. The refore, the larger the we ights ar e, the mo re the smoothed values follow the data; the smaller the weights are, the smoother the pattern

in the smoothed values is.

1 28 CHAPTER 4 Moving Averages and Smoothing Methods

We could develop a grid of values of a and /3 ( e.g., each combination of a = 0.1, 0.2, ... , 0.9 and f3 = 0.1, 0.2, ... , 0.9) and then select the combination that provides the lowest MSE. Most forecasting sof_tware packages use an o?t~rnization algorithm to minimize the MSE. We might insist that a = /3 ,_thus prov1dmg equal amounts of smoothing for the level and the trend. In the special case where a = /3, Holt's approach is the same as Brown's double exponential smoothing.

To get started, initial values for L and Tin Equations 4.15 and 4.16 must be deter- mined. One approach is to set the first estimate of the smoothed level equal to the first observation. The trend is then estimated to be zero. A second approach is to use the aver- age of the first five or six observations as the initial smoothed value L. The trend is then estimated using the slope of a line fit to these five or six observations. Mini tab develops a regression equation using the variable of interest as Y and time as the independent vari- able X The constant from this equation is the initial estimate of the level component, and the slope or regression coefficient is the initial estimate of the trend component.

Example 4.9 In Example 4.6, simple exponential smoothing did not produce successful forecasts of Acme Tool Company saw sales. Because Figure 4-8 suggests that there might be a trend in these data, Holt's linear exponential smoothing is used to de velop forecasts. To begin the computa- tions shown in Table 4-8, two estimated initial values are needed, name ly, the initial level and the initial trend value. The e stimate of the level is set equal to the first observation. Toe trend is estimated to equal zero. The technique is demonstrated in Table 4-8 for a = .3 and /3 = .1.

The value for a used here is close to the optimal a ( a = .266) for simple exponential smoothing in Example 4.6. a is used to smooth the data to eliminate randomness and esti• mate level. The smoothing constant /3 is like a , except that it is used to smooth the trend in the data. Both smoothing constants are used to average past values and thus to remove ran· domness. The computations leading to the forecast for period 3 are shown next.

1. Update the exponentially smoothed series or level:

L, = aY, + (1 - a )(L ,-1 + Ti- 1) L 2 = .3Yi + (1 - .3 )(L 2-1 + Tz-1) L 2 = .3 (350 ) + .7(500 + 0 ) = 455

2. Update the trend estimate:

7; = 13 (L , - L ,-1) + (1 - 13 ) 7;-1 Tz = .l(L2 - L 2- 1) + (1 - .l) Tz- 1 Tz = .1 (455 - 500) + .9(0 ) = -4.5

3. Forecast one period into the future:

Y,+p = L , + pT,

Yz+i = Lz + lTz = 455 + 1 (-4.5 ) = 450.5

4. D e termine the forecast error:

e3 = ~ - Yj = 250-450.5 = -200.5

The forecast for period 25 is computed as follows:

1. Update the exponentially smoothed series o r level:

Lz4 = .3Y24 + (1 - .3)( Lz4-1 + Tz4-1) L 24 = .3(650) + .7 (517.6 + 9.8) = 564.2

CHAPTER 4 Moving Averages and Smoothing Methods 1 29

TABLE 4-8 Exp t· 11 S onen 1a y moothed Values for Acme Tool Company Sales, Holt's Method, for Example 4.9

Year t Y, L, T, yt+p e, 2000 1 500 500.0 0.0 500.0 0.0

2 350 455.0 -4.5 500.0 -150.0

3 250 390.4 -10.5 450.5 -200.5 4 400 385.9 -9.9 379.8 20.2

2001 5 450 398.2 -7.7 376.0 74.0

6 350 378.3 -8.9 390.5 -40.5

7 200 318.6 -14.0 369.4 -169.4

8 300 303.2 - 14.1 304.6 -4.6

2002 9 350 307.4 -12.3 289.1 60.9

10 200 266.6 -15.2 295.1 -95.1

11 150 221.0 - 18.2 251.4 -101.4

12 400 262.0 -12.3 202.8 197.2

2003 13 550 339.8 -3.3 249.7 300.3

14 350 340.6 -2.9 336.5 13.5

15 250 311.4 -5.5 337.7 -87.7

16 550 379.1 1.8 305.9 244.1

2004 17 550 431.7 6.9 381.0 169.0

18 400 427.0 5.7 438.6 - 38.6

19 350 407.9 3.3 432.7 -82.7

20 600 467.8 8.9 411.2 188.8

2005 21 750 558.7 17.1 476.8 273.2

22 500 553.1 14.8 575.9 -75.9

23 400 517.6 9.8 567.9 -167.9

24 650 564.2 13.5 527.4 122.6

2006 25 850 577.7 272.3

MSE = 20,515.5

2. Update the trend estimate:

Tz4 = .l(L24 - ~ -d + (1 - .l)Tz4- 1 Tz4 = .1(564.2 - 517.6) + .9(9.8) = 13.5

3. Forecast one period into the future:

½4+1 = L24 + l'.fz4 Yis = 564.2 + 1(13.5) = 577.7

On the basis of minimizing the MSE over the period 2000 to 2006, Holt's linear smoothing (with a = .3 and f3 = .1) does not_reproduce the data any bette~ than sim- ple exponential smoothing that used a smoothmg constant of .266. A companson of the MAP Es shows them to be about the same. When the forecasts for the actual sales for

1 3 0 CHAPTER 4 Moving Averages and Smoothing Methods

Holt's Linear Exponential Smoothing Plot for Saws

800

700

600

~ 500

(11

Cll 400

300

200

Minitab Instructions Stat>Time Series>Double Exp. Smootning

,------, Variable

--- Actual

-- Fb Smoothing Coratws Alpha (level) 03 Garron>t (tr<nd) 0 ,1

Al:.ctX ¥:.Y Musures MAPE 35,4

MAO 1253 MSO 20515,5

lOOL-.------.-.----,-.------.-.----,---.-,r--::,:---::~ 2 4 6 8 ID ~ M ~ IB 20 ~ ~

Quarter

FIGURE 4 - 1 o Holt's Linear Exponential Smoothing for Acme Tool Company Data for Example 4 .9

the first quarter of 2006 are compared, again Holt smoothing and simple exponential smoothing are comparable. To summarize:

a= .266 a = .3, /3 = .l

MSE = 19,447 MSE = 20,516

MAPE = 32.2% MAPE = 35.4%

Figure 4-10 shows the results when Holt's method using a = .3 and {3 = .l is run on Minitab.2 The autocorrelation function for the residuals from Holt's linear expo- nential smoothing is given in Figure 4-11. The autocorre lation coefficients at time lags 2 and 4 appear to be significant. Also, when the Ljung-Box Q statistic is computed for six time lags, the large value of LBQ (36.33) shows that the residuals contain extensive autocorre lation; they are not random. The large residual autocorrelations at lags 2 and 4 suggest a seasonal component may be present in Acme Tool Company data.

The results in Examples 4.6 and 4.9 (see Figures 4-8 and 4-10) a re not much differ- ent because the smoothing constant a is about the sam e in both cases and the smooth· ing constant f3 in Example 4.9 is small. (For f3 = 0, Holt's linear smoothing reduces to simple exponential smoothing.)

Exponential Smoothing Adjusted for Trend and Seasonal Variation: Winters' Method Examination of the data for Acme Tool Company in Table 4-8 indicates that sales are consistently higher during the first and fourth quarters and lower during the third quarter. A seasonal pattern appears to exist. Winters' three-parameter linear and sea· sonal exponential smoothing method, an extension of Holt's method, might repres~nt the data better and reduce forecast error. In Winters' m ethod, one additional equation is used to estimate seasonality. In the multiplicative version of Winters' method, the seasonality estimate is given as a seasonal inde x , and it is calculated with Equation 4.20. Equation 4.20 shows that to compute the current seasonal component, Sr, the

2 In the Minitab program, the trend parameter gamma (y) is identical to our beta (/3) .

CHAPTER 4 Moving Averages and Smoothing Methods 1 3 1

Autocorrelation Function: Residuals

Laq ACF LBQ

1.0

0.8

Autocorttlation function few Example 4.9 Residuals l 0.092547 2 -0.629408 3 0 .074355 4 0.601371 5 -0.070132 6 -0.618267

T o. 45

- 3 .06 0 . 27 2.18

- 0.22 - 1. 90

0. 23 11.47 11.63 22.92 23 . 08 36.33 i li.µ..1_-__ -~---_-_-~,_-__ -T_,_-_-_-_--r-------r,

¥ ·0.2 J ·0,4 ---- ---... .. ______ _

..0.6 ---------- -0.8

-1.0

FIGURE 4 - 11 Autocorrelation Function for the Residuals from Holt's Linear Exponential Smoothing for A cme Tool Company Data for · Example 4 .9

product of 'Y and an estimate of the seasonal index given by Yr / L, is added to ( 1 - y ) times the previous seasonal component, S,-s· This procedure is equivalent to smoothing current and previous values of Yr/ L ,. Yr is divided by the current level estimate, L,, to create an index (ratio) that can be used in a multiplicative fashion to adjust a forecast to account for seasonal peaks and valleys.

The four equations used in Winters' (multiplicative) smoothing are

1. The exponentially smoothed series, or level estimate:

Yr ) L, = a-+ (l - a )( L,- 1 + Tr-1 s,-s

2. The trend estimate:

3. The seasonality estimate:

Yr ) s, = -y - + (1 - -y s,-s L,

4. Toe forecast for p periods into the future:

'Yc+p = (L , + pT;)S,-s+p

where L = the new smoothed value or current level estimate

I

a = the smoothing constant for the level Y. = the new observation or actual value in period t

I

/3 = the smoothing constant for the trend estimate T. = the trend estimate

I

'Y = the smoothing constant for the seasonality estimate s, = the seasonal estimate

(4.18)

(4.19)

(4.20)

(4.21)

1 3 2 CHAPTER 4 Moving Averages and Smoothing Methods

p = the periods to be forecast into the future s = the length of seasonality

Yr+p = the forecast for p periods into the future Equation 4.18 updates the smoothed series. A slight difference in this equation distin- guishes it from the corresponding one in Holt's procedure, Equation 4.15. In Equation 4.18, Y, is divided by S,-s, which adjusts Y, for seasonality, thus removing the seasonal effects that might exist in the original data, Y,.

After the trend estimate and seasonal estimate have been smoothed in Equations 4.19 and 4.20, a forecast is obtained with Equation 4.21. lt is almost the same as the cor- responding formula, Equation 4.17, used to obtain a forecast with Holt's smoothing. The difference is that this estimate for future periods, t + p, is multiplied by S,-s+p· This seasonal index is the last one available and is therefore used to adjust the forecast for

seasonality. As with Holt's linear exponential smoothing, the weights a, {3, and -y can be

selected subjectively or generated by minimizing a measure of forecast error, such as the MSE. The most common approach for determining these values is to use an opti- mization algorithm to find the optimal smoothing constants.

To begin the algorithm for Equation 4.18, the initial values for the smoothed series, L,; the trend , 7;; and the seasonal indices, S1, must be set. One approach is to set the first estimate of the smoothed series (level) equal to the first observation. The trend is then estimated to equal zero, and each seasonal index is set to 1.0. Other approaches for ini- tializing the level, trend , and seasonal estimates are available. Minitab, for example. develops a regression equation using the variable of interest as Y and time as the inde- pendent variable X. The constant from this equation is the initial estimate of the smoothed series or level component, and the slope or regression coefficient is the ini- tial estimate of the trend component. Initial values for the seasonal components are obtained from a dummy variable regression using detrended data (see Chapter 8).

Example 4.10 Winters' technique is demonstrated in Table 4-9 for a = .4, (3 = .1 , and 'Y = .3 for the Acme Tool C?m~any data. The valu~ f_or a is similar to the one used for simple exponential smoothing m Example 4.6, and 1t 1s used to smooth the data to create a level estimate. The smoot~g constant /3 is used to create_ a smoothed estimate of trend. The smoothing con- stant 'Y _1s_ used to create a smoothed ~st1mate of the seasonal component in the data.

Mllltab can be used t? solve _t his example (see the Minitab Applications section at the end of the chapter for_ the mstructlons).3 1:'he results are shown in Table 4-9 and Figure 4-12. The forecast for the first quarter of 2006 1s 778.2. The computations leading to the forecast value for the first quarter of 2006, or period 25, are shown next.

1. The exponentially smoothed series, or level estimate:

Y, L 1 = ~ + (1 - a)(L,_1 + T,_i)

1-s

">14 L24 = .4~ + (1 - .4)(L24-1 + Ti4 -i)

"24-4 650

L 24 = ·\.39628 + (1 - .4)(501.286 + 9.148)

L24 = .4(465.52) + .6(510.434) = 492.469

31n the Minitab program, the trend parameter gamma (-y) is identical to our beta ra) d th 1 am • • . . . v, an e seasona ar - eter delta (o) 1s 1dent1cal to our gamma (y) m Equations 4.19 and 4 20 res u· 1 p . , pee ve y.

2.

3.

CHAPTER 4 Moving Averages and Smoothing Methods 1 3 3

TABLE 4-9 Exponentially Smoothed Values for Acme Tool Company Sales, Winters' Method, for Example 4. 10

Year t Y, L, T, s, Yt+p e,

2000 1 500 415.459 -41.9541 1.26744 563.257 -63.257

2 350 383.109 -40.9937 0.89040 328.859 21.141

3 250 358.984 -39.3068 0.66431 222.565 27.435

4 400 328.077 -38.4668 1.18766 375.344 24.656

2001 5 450 315.785 -35.8494 1.31471 367.063 82.937

6 350 325.194 -31.3235 0.94617 249.255 100.745

7 200 296.748 - 31.0358 0.66721 195.221 4.779

8 300 260.466 -31.5604 1.17690 315.576 - 15.576

2002 9 350 243.831 -30.0679 1.35093 300.945 49.055

10 200 212.809 -30.1632 0.94426 202.255 -2.255

11 150 199.515 -28.4764 0.69259 121.863 28.137

12 400 238.574 -21.7228 1.32682 201.294 198.706

2003 13 550 292.962 - 14.1117 1.50886 292.949 257.051

14 350 315.575 -10.4393 0.99371 263.306 86.694

15 250 327.466 -8.2062 0.71385 211.335 38.665

16 550 357.366 -4.3956 1.39048 423.599 126.401

2004 17 550 357.588 -3.9339 1.51763 532.584 17.416

18 400 373.206 -1.9787 1.01713 351.428 48.572

19 350 418.856 2.7843 0.75038 264.999 85.001

20 600 425.586 3.1788 1.39628 586.284 13.716

2005 21 750 454.936 5.7959 1.55691 650.706 99.294

22 500 473.070 7.0297 1.02907 468.626 31.374

23 400 501.286 9.1484 0.76465 360.255 39.745

24 650 492.469 7.3518 1.37336 712.712 -62.712

2006 25 850 778.179

26 600 521.917

27 450 393.430

28 700 716.726

MSE = 7,636.86

The trend estimate: T, == {3(L1 - L,-1 ) + (1 - /3 ) T,- 1

7;4 = .l(L24 - Li4-i) + (1 - .l)'fi4-1 'fi4 = .1( 492.469 - 501.286) + .9(9.148) 7i.4 == .1(-8.817) + .9(9.148) = 7.352

Toe seasonality estimate: Y, ) S, = 'Y - + (1 - 'Y S,-s L, )'i4

.5i4 = .3- + ( 1 - .3)S24-4 L24

650 s

24 = .3

492 .4

69 + .7(1.39628)

S24 = .3(1.3199) + .9774 = 1.3734

134 CHAPTER 4 Moving Averages and Smoothing Methods

800

700

600

II) 500

3t (0 (J'J 400

300

200

100

~

Winters' Method Plot for Saws Multiplicative Model

Minitab Instructions Stat> Time Series>Winters' Metnod

2 4 6 8 ill ~ M ffi IB ~ ~ ~ Quarter

V•riable - Actual

---- Fils

Smoothing Consl•nls Alpha (level) 0 .4 Gamma ( trend) 0. 1 Delta (seasonal) 0 3

Accuracy Measures MAPE 15.21 MAD &355 MSD 7&3& .8&

FIGURE 4-12 Winters' Exponential Smoothing for Acme Tool Company Data for Example 4. 10

4. The forecast for p = 1 period into the future:

Yi4+1 = ( L14 + l T24)S24- 4+1 ¥'25 = (492.469 + 1 (7.352))1.5569 = 778.17

. For the p~rameter valu es_ considered, Winters' technique is better than both of the pre- v10us smoothing pr?cedures m terms of minimizing the MSE. When the forecasts for the actual sa~es for_ the first quarter of 2006 are compared , Winters' technique also appears to do a better JOb. ~1gure 4-13 shows the ~utocorrelatio n function for the Winters' exponential smoothing residuals. None of the res~dual autocorre lation coefficients appears to be signifi- cantly larger than zero. When the LJung-Box Q statistic is calculated for six time lags, the small v~lue of LBQ (5 .01) shows that the residual series is random. Winters' exponential smoothmg method seems to provide adequate forecasts for the Acme Tool Company data.

Winters' method provide~ an easy way to account for seasonality when data have a se~so~al pattern. An alternative _method consists of first deseasonalizing or seasonally adJustmg the data. Desea~onalizmg is a process that removes the effects of seasonality from the raw data and will be demonstrate d in C h apter 5 Th f t' d I · . . . e orecas mg mo e 1s applied to the deseasonahzed data, and if required th e season al t · · . , componen 1s rem- serted to provide accurate forecasts.

Exponential smoothing is a popular technique for short f · I · d • . . -run orecastlllg. ts maior a vantages are its low cost and Slillplicity When forecasts are d d f · . . . · nee e or mventory sys- tems contammg thousa nds of items, smoothing methods are often the only acceptable approach.

Simple moving averages and expone f 1 · f

n ia smoothing base forecasts on weiohted averages o past measurements. The rationale is that past v 1 · ·ru O · about what will occur in th fut s· . a ues contam I ormat1on well as information concer~ t::n mce _past values mclud~ random fluctua~ions as

g derlymg pattern of a vanable, an attempt 1s made

CHAPTER 4 Moving Averages and Smoothing Methods 1 3 5

Autocorrelation Function: Residuals

Lag ACF T LBQ l 0 .297749 1.46 2 .41 2 -0.031543 -0 .14 2 . 43 3 0 . 01 7610 0.08 2.44 4 -0.107084 -0.48 2 . 80 5 -0 . 25 7066 -1.15 4 . 97 6 0 . 033280 0.14 5.01

1,0 0.8

0.6

Autocorrel~tktn Functio" rer Winlffs' ResHkl•ls

g 0,4 _ __ .,.. _____ _ _____ _ __ _ _

i 0 ~ I ~ 0,0t-"---~--~~--..--, --~ 1 --.......J ~ -0.2

l -o.4 -0.&

-0.8

-1.0

4 S

FIGURE 4-1 3 Autocorrelation Function for the Residuals from Winters' Multiplicative Exponential Smoothing Method for Acme Tool Company Data for Example 4. 1 O

to smooth these valu es. Smoothing assumes that extreme fluctuations represe nt ran- domness in a series of historical observations.

Moving averages are the means of a certain number, k, of values of a variable. The most recent average is then the forecast for th e next period. This approach assigns an equal weight to each past value involved in th e average. However, a convincing argu- ment can b e made for using all t he data but emphasizing the most recent values. Exponential smoothin g methods ar e attractive because they ge nerate forecasts by u sing all the o bse rvations and assigning weights that decline exponentiall y as the observations get older.

APPLICATION TO MANAGEMENT

Forecasts are one of the most impor tant inpu ts managers have to aid them in the d ecision-making process. Virtually every impo rtant operating decision depends to some extent on a forecast . The production department has to schedule employment need s and raw material orders for th e next month or two; the finance department must determine the best investment opportunities; marketing must forecast the demand for a n ew product. Toe list of forecasting applications is quite lengthy.

Executives are keenly aware of the importance of forecasting. Indeed, a great deal of time is spent studying trends in economic and political affairs and how events might affect demand for products and/or services. One issue of interest is the importance executives place on qua ntitative forecasting methods compa~e_d to t?eir own opinions. This issue is especially sensitive when e_ve~ts that hav~ a significant_ lillpact on demand are involved. One problem with quantitative forecastmg methods 1s that they depend on historical data. For this reason, they are probably least effective in determining the dramatic change that often r esults in sharp!~ higher or low~r dema~d- .

The averaging and smoothi~g f~r~castmg m~thods discussed m this chapter are useful because of their relative sunphc1ty. These Slillple methods tend to be less costly, easier to implement, and easier to ~nderst~nd than co°:1plex methods: F~equently, the cost and potential difficulties associated with constructlllg more so_phisttcat_ed m_odels outweigh any gains in their accuracy. For these reasons, small businesses fmd simple

136 CHAPTER 4 Moving Averages and Smoo thing Methods

Glossary

me thods practical. Businesses without personnel capable of handling statistical models also tum to simple methods. Business managers frequently face the need to prepare short-term forecasts for a number of different items. A typical example is the manager who must schedule production on the basis of some forecast of demand for several hundred different products in a product line. Also, new businesses without lengthy his- torical databases find these simple approaches helpful.

W ith a judicious choice of order k , the moving average method can do a good job of adjusting to shifts in levels. It is economical to update, and it does not require much data storage. The moving average method is most frequently used when repeated fore-

casts are necessary. Exponential smoothing is a popular technique whose strength lies in good short-

te rm accuracy combined with quick, low-cost updating. The technique is widely used when regular monthly or weekly forecasts are needed for a large number, perhaps thousands, of items. Inventory control is one example where exponential smoothing methods are routinely used.

Exponential smoothing. Exponential smoothing is a procedure for continually revising a forecast in the light of more-recent experience.

Moving average. A moving average of order k is the mean value of k consecutive observations. The most recent moving average value provides a fore- cast for the next period.

Simple average. A simple average uses the mean of all relevant historical observations as the fore- cast for the next period.

Tracking signal. A tracking signal involves com- puting a measure of forecast errors over time and setting limits so that, when the cumulative error goes outside those limits, the forecaster is alerted.

Key Formulas

Naive model

Yi+1 = Y, Naive trend model

Naive rate of change model

Naive seasonal model for quarterly data

Yi+t = Y,_3 Naive trend and seasonal model for quarterly data

Simple average model

A Y, - Y,- 4 Yr+1 = Yr-3 + __ __.:,__;_

4

,.. 1 I Yr+1 = - LY;

t i = l

(4.1)

(4.2)

(4.3)

(4.4)

(4.5)

(4.6)

CHAPTE R 4 M oving Averages and Smoothing Methods 13 7

Updated simple average, new period

y; _ tYi+1 + Yr+t ,+2 - t + l

Moving average for k time periods

Y. A Y, + Y,-1 + ... + Y,- k+ I t + l =

k

Double moving average

M ' = M, + M ,-1 + M ,- 2 + .. . + M ,- k+ I I k

a,= 2M, - M; 2

b, = k _ l ( M, - M ;)

Yi+p =a,+ b,p

Simple exponential smoothing

Yi+ 1 = aY, + ( 1 - a)Y,

Equivale nt alternative expression : A 2 3 Y,+1 = aY, + a(l - a ) Y,- 1 + a (l - a ) Yr- 2 + a(l - a) Yr- 3 + ·· ·

Holt's linear smoothing The exponentially smoothed series, or current level estimate:

L , = a Y, + (1 - a )(L ,-1 + Tr-1)

The tre nd e stimate:

The forecast for p periods into the future:

Y,+p = L , + pT,

Winters' multiplicative smoothing The exponentially smoothed series, or le ve l estimate:

The trend es timate:

The se asonality e stimate:

L , = a Y, + (1 - a)( L ,-1 + T,-1) S,- s

The forecast for p periods into the future:

Y,+p = (L, + pT,)S,-s+p

(4.7)

(4.8)

(4.9)

(4.10)

(4.11)

(4.12)

(4.13)

(4.14)

(4.15)

(4.16)

(4.17)

(4.18)

(4.19)

(4.20)

(4.21)

1 38 CHAPTER 4 Moving Averages and Smoothing Methods

Problems 1. Which forecasting technique continually revises an estimate in the light of more-

recent experiences?

2. Which forecasting technique uses the value for the current period as the forecast for the next period?

3. Which forecasting technique assigns equal weight to each observation?

4. Which forecasting technique(s) should be tried if the data are trending?

5. Which forecasting technique(s) should be tried if the data are seasonal?

6. Apex Mutual Fund invests primarily in technology stocks. The price of the fund at the end of each month for the 12 months of 2006 is shown in Table P-6. a. Find the forecast value of the mutual fund for each month by using a naive

model (see Equation 4.1). The value for December 2005 was 19.00. b. Evaluate this forecasting method using the MAD. c. Evaluate this forecasting method using the MSE. d. Evaluate this forecasting method using the MAPE. e. Evaluate this forecasting method using the MPE. f. Using a naive model, forecast the mutual fund price for January 2007. g. Write a memo summarizing your findings.

7. Refer to Problem 6. Use a three-month moving average to forecast the mutual fund price for January 2007. ls this forecast better than the forecast made using the naive model? Explain.

8. Given the series Y, in Table P-8: a. What is the forecast for period 9, using a five-period moving average? b. If simple exponential smoothing with a smoothing constant of .4 is used, what is

the forecast for time period 4? c. In part b, what is the forecast error for time period 3?

9. The yield on a general obligation bond for the city of Davenport fluctuates with the market. The monthly quotations for 2006 are given in Table P-9.

TAB LE P-6

Month

January February March April May June July August September October November December

Mutual Fund Price

19.39 18.96 18.20 17.89 18.43 19.98 19.51 20.63 19.78 21.25 21.18 22.14

CHAPTER 4 Moving Averages and Smoothing Methods 1 39

TABLE P-8

Time Period Y, Y, e,

1 200 200 2 210 3 215 4 216 5 219 6 220 7 225 8 226

TABLE P-9

Month Yield

January 9.29 Februar y 9.99 March 10.16 April 10.25 May 10.61 June 11.07 July 11.52 August 11.09 September 10.80 October 10.50 November 10.86 December 9.97

a. Find the forecast value of th e yield for the ob ligation bonds for each month, starting with April, using a three-month moving average.

b. Find the forecast value of the yield for the obligation bonds for each month , starting with June, using a five-month moving average.

c. Evaluate these forecasting methods using the MAD. d. Evaluate these forecasting methods using the MSE. e. Evaluate these forecasting methods using the MAPE. f. E valuate these forecasting methods using the MPE. g. Forecast the yield for January 2007_ us~g the better technique. h. Write a m emo summarizing your fmdmgs.

10. This question refers to Problem 9. Use exponential srno~thing with a smoothing constant of .2 and an initial value of 9.29 to forecast the yield for Januar y 2007. Is this forecast b etter than the forecast made using the better moving average

mo del? Explain. 11. Toe Hughes Supply Company uses an inventory management method to deter-

mine the monthly d emands for various products. The demand values for the last 12 onths of e ach product have been recorded and are available for future fo recast-

: g. The d em and values for the 12 months of 2006 for one electrical fixture are pre-

sented in Table P-11.

140 CHAPTER 4 Mo ving Averages and Smoothing Methods

TABLE P-11

Month Demand

January 205

February 251

March 304

April 284

May 352

June 300

July 241

August 284

September 312

October 289

November 385

D ecember 256

Source-· Based on Hughes Supply Company records.

Use exponential smoothing with a smoothing constant of .5 and an initial value of 205 to forecast the demand for January 2007.

12. General American Investors Company, a closed-end regulated investment man- agement company, invests primarily in m edium- and high-q uality stocks. Jim Campbell is studying the asset value per share for this company and would like to forecast this variable for the remaining quarters of 1996. The data are presented in Table P-12 .

Evaluate the ability to forecast the asset value per share variable of the follow- ing forecasting methods: naive, moving average, and exponential smoothing. When you compare techniques, take into consideration that the actual asset value per share for the second quarter of 1996 was 26.47. Write a report for Jim indicating which method he should use and why.

TABLE P-1 2 General American Investors Company Assets per Share, 1 985-1 996

Quarter

Year I 2 3 4

1985 16.98 18.47 17.63 20.65 1986 21.95 23.85 20.44 19.29 1987 22.75 23 .94 24.84 16.70 1988 18.04 19.19 18.97 17.03 1989 18.23 19.80 22.89 21.41 1990 21.50 25.05 20.33 20.60 1991 25.33 26.06 28.89 30.60 1992 27.44 26.69 28.71 28.56 1993 25.87 24.96 27.61 24.75 1994 23.32 22.61 24.08 22.31 1995 22.67 23.52 25.41 23.94 1996 25.68

Source: The Value Line In vestment Survey (Ne y k· V 1 L' 1990, 1993, 1996). w or . a ue me,

CHAPTER 4 Moving Averages and Smoothing Methods 14 1

TABLE P- l 3 Southdown Revenues, 1 986-1999

Quarter

Year I 2 3 4

1986 77.4 88.8 92.1 79.8 1987 77 .5 89.1 92.4 80.1 1988 74.7 185.2 162.4 178.1 1989 129.1 158.4 160.6 138.7 1990 127.2 149.8 151.7 132.9 1991 103.0 136.8 141.3 123.5 1992 107.3 136.1 138.6 123.7 1993 106.1 144.4 156.1 138.2 1994 111.8 149.8 158.5 141.8 1995 119.1 158.0 170.4 151.8 1996 127.4 178.2 189.3 169.S 1997 151.4 187.2 199.2 181.4

1998 224.9 317.7 341.4 300.7

1999 244.9 333.4 370.0 326.7

Source: The Value Line Investment Survey (New York: Value Line, 1990, 1993, 1996, 1999).

13. Southdown, Inc. , the nation's third largest cement producer, is pushing ahead with a waste fuel burning program. The cost for Southdown will total about $37 million. For this reason, it is extremely important for the company to have an accurate forecast of revenues for the first quarter of 2000. The data are presented

14.

15.

16.

in Table P-13. a. Use exponential smoothing with a smoothing constant of .4 and an initial value

of 77.4 to forecast the quarterly revenues for the first quarter of 2000. b. Now use a smoothing constant of .6 and an initial value of 77.4 to forecast the

quarterly revenues for the first quarter of 2000. c. Which smoothing constant provides the better forecast? d. Refer to part c. Examine the residual autocorrelations. Are you happy with sim-

p le exponential smoothing for this example? Explain.

The Triton Energy Corporation explores for and produces oil and gas. Company president Gail Freeman wants to have her company's analyst forecast the com- pany's sales per share for 2000. This will be an important forecast, since Triton's restructuring plans have hit a snag. The data are presented in Table P-14.

D etermine the best forecasting method and forecast sales per share for 2000.

Tue Consolidated Edison Company sells electricity (82% of revenues), gas (13%), and steam (5%) in New York City and Westchester County. Bart Thomas, company forecaster, is assigned the task of forecasting the company's_quarterly revenues for the rest of 2002 and all of 2003. He collects the data shown in Table P-15.

D etermine the best forecasting technique and forecast quarterly revenue for

the rest of 2002 and all of 2003. A job-shop manufacturer that specializes in replacement parts bas no forecasting system in place and manufactures products based on last month's sales. Twenty- four months of sales data are available and are given in Table P-16.

142 CHAPTER 4 Moving Averages and Smoothing Methods

TABLE P-14 Triton Sales per Share, 1974-1 999

Year Sales per Share Year Sales per Share

1974 .93 1987 5.33

1975 1.35 1988 8.12

1976 1.48 1989 10.65

1977 2.36 1990 12.06

1978 2.45 1991 11.63

1979 2.52 1992 6.58

1980 2.81 1993 2.96

1981 3.82 1994 1.58

1982 5.54 1995 2.99

1983 7.16 1996 3.69

1984 1.93 1997 3.98

1985 5.17 1998 4.39

1986 7.72 1999 6.85

Source: The Value Line In vestment Survey ( ew York: Value Line,

1990,1993, 1996,1999)

TABLE P- 1 5 Quarterly Revenues for Consolidated Edison (S millions), 1 985- June 2002

Year Mar. 31 Jun. 30 Sept. 30 Dec. 31

1985 1,441 1,209 1,526 1,321 1986 1,414 1,187 1,411 1,185 1987 1,284 1,125 1,493 1,192 1988 1,327 1,102 1,469 1,213 1989 1,387 1,218 1.575 1,371 1990 1,494 1.263 1,613 1,369 1991 1,479 1,330 1,720 1,344 1992 1,456 1,280 1,717 1,480 1993 1,586 1,396 1,800 1,483 1994 1,697 1,392 1,822 1,461 1995 1,669 1,460 1,880 1,528 1996 1,867 1,540 1,920 1,632 1997 1,886 1,504 2 ,011 1,720 1998 1,853 1,561 2,062 1,617 1999 1,777 1,479 2,346 1,889 2000 2,318 2,042 2,821 2,250 2001 2,886 2,112 2,693 1,943 2002 2,099 1,900

Source: The Value Line lnvesrment Survey (New York: Value L. 1990 1993 19%,1999,2001). me, - ,

a. Plot the sales data as a time series Are the data l? H . · seasona . mt: For monthly data, _the seasonal period is s = 12. Is there a pattern (e.g ..

~ummer sales relatively low, fall sales relatively high) that tends to repeat itself every 12 months?

17.

18.

CHAPTER 4 Moving Averages and Smoothing Methods 143

TABLE P-16

Month Sales Month Sales

January 2005 430 January 2006 442 February 420 February 449 March 436 March 458 April 452 April 472 May 477 May 463 June 420 June 431 July 398 July 400 August 501 August 487 September 514 September 503 October 532 October 503 November 512 November 548 December 410 December 432

b. Use a naive model to generate monthly sales forecasts (e.g. , the February 2005 forecast is given by the January 2005 value, and so forth). Compute the MAPE.

c. Use simple exponential smoothing with a smoothing constant of .5 and an initial smoothed value of 430 to generate sales forecasts for each month. Compute the

MAPE. d. Do you think either of the models in parts band c is likely to generate accurate

forecasts for future monthly sales? Explain. e. Use Minitab and Winters' multiplicative smoothing method with smoothing con-

stants a = {3 = -y = .5 to generate a forecast for January 2007. Save the residuals. f. Refer to part e. Compare the MAP£ for Winters' method from the computer

printout with the MAP Es in parts band c. Which of the three forecasting proce-

dures do you prefer? g. Refer to part e. Compute the autocorrelations (for six lags) for the residuals

from Winters' multiplicative procedure. Do the residual autocorrelations sug- gest that Winters' procedure works well for these data? Explain.

Consider the gasoline purchases for the Spokane Transit Authority given in Table 4-2. In Example 4.3, a five-week moving average is used to smooth the data and

generate forecasts. a. Use Minitab to smooth the Spokane Transit Authority data using a four-week

moving average. Which moving average length, four weeks or five weeks, appears to represent the data better? Explain.

b. Use Mini tab to smooth the Spokane Transit Authority data using simple expo- nential smoothing. Compare your results with those in part a. Which procedure, four-week moving average or simple exponential smoothing, do you prefer for

these data? Explain. Table P-18 contains the number of severe earthquakes (those with a Richter scale magnitude of 7 and above) per year for the years 1900-1999. a. u e Minitab to smooth the earthquake data with moving averages of orders

k ~ 5 10 and 15. Describe the nature of the smoothing as the order of the mov- ing av~ra~e increases. Do you thi~ there might be a cycle in these data? If so, provide an estimate of the length (m years) of the cycle.

1 44 CHAPTE R 4 Moving Averages and Smoothing Meth ods

TABLE P-18 Number of Severe Earthquakes, 1 900-1 999

Year N umber Year N umber Year N umber Year N umber

1900 13 1927 20 1954 17 1981 14

1901 14 1928 22 1955 19 1982 10

1902 8 1929 19 1956 15 1983 15

1903 10 1930 13 1957 34 1984 8

1904 16 1931 26 1958 10 1985 15

1905 26 1932 13 1959 15 1986 6

1906 32 1933 14 1960 22 1987 11

1907 27 1934 22 1961 18 1988 8

1908 18 1935 24 1962 15 1989 7

1909 32 1936 21 1963 20 1990 12

1910 36 1937 22 1964 15 1991 11

1911 24 1938 26 1965 22 1992 23

1912 22 1939 21 1966 19 1993 16

1913 23 1940 23 1967 16 1994 15

1914 22 1941 24 1968 30 1995 25

1915 18 1942 27 1969 27 1996 22

1916 25 1943 41 1970 29 1997 20

1917 21 1944 31 1971 23 1998 16

1918 21 1945 27 1972 20 1999 23

1919 14 1946 35 1973 16 1920 8 1947 26 1974 21 1921 11 1948 28 1975 21 1922 14 1949 36 1976 25 1923 23 1950 39 1977 16 1924 18 1951 21 1978 18 1925 17 1952 17 1979 15 1926 19 1953 22 1980 18

Source: U.S. Geological Survey Earthquake Hazard Program .

b. Use Minitab to smooth the earthquake d ata us ing simple exponential smooth- ing. Store the residuals and generate a forecast for the number of severe earth- quakes in the year 2000. Compute the residual a utocorrelations. Does simple expo nential smoothing provide a reasonable fit to these data? Explain.

c. l s there a seasonal component in the e ar thq uake d ata? Why or why not?

19. Table P-23 in Chapter 3 contains the quarterly income before extraordinary items for Southwest Airlines for the years 1988-1999. a. Using ~ nita b, smooth the ~outhwest A irlines income data using Holt's linear

~mootbing a~d store the residuals. Compute the residual a utocorrelations. Does 1t appear as tf Holt's smoothing proced ure represents these data well? If not, what time series component (tre nd, cycle, seasonal) is no t accounted for by H olt's method?

b. l!s~ ~ta b to smooth the So uthwest Airlines income data using Wmters' mul- !1plicative exponential smoothing. Store the residuals and generate forecasts of mcome ~or the four quart ers of 2000. Compute the residual autocorrelations. Does Wmters' smoo~hing techniq ue fit the income data well? Do the forecasts seem reasonable? Discuss.

CASES

CAS E 4-1

CH A PTE R 4 M oving Averages and Smoothing Methods 145

TABLE P-20 Quarterly Sales for The Gap, Fiscal Years 1980-2004

Year QI Q2 Q3 Q4

1980 65.3 72.1 107.1 125.3 1981 76.0 76.9 125.4 139.1 1982 84.4 86.0 125.4 139.1 1983 87.3 90.9 133.3 169.0 1984 98.0 105.3 141.4 189.4 1985 105.7 120.1 181.7 239.8 1986 160.0 164.8 224.8 298.4 1987 211.1 217.7 273.6 359.6 1988 241.3 264.3 322.8 423.7 1989 309.9 325.9 405.6 545.2

1990 402.4 405.0 501.7 624.7

1991 490.3 523.1 702.0 803.5

1992 588.9 614.1 827.2 930.2

1993 643.6 693.2 898.7 1,060.2

1994 751.7 773 .1 988.3 1,209.8

1995 848.7 868.5 1,155.9 1,522.1

1996 1,113.2 1,120.3 1,383.0 1,667.9

1997 1,231.2 1,345.2 1,765.9 2,165.5

1998 1,719.7 1,905.0 2,399.9 3,029.8

1999 2,277.7 2,453.3 3,045.4 3,858.9

2000 2,732.0 2,947.7 3,414.7 4,579.1

2001 3,179.7 3,245.2 3,333.4 4,089.6

2002 2,890.8 3,268.3 3,645.0 4,650.6

2003 3,353.0 3,685.0 3,929.0 4,887.0

2004 3,668.0 3,721.0 3,980.0 4,898.0

So urce: Based on The Value L ine Investment Survey (New York: Value Line, various years), and lOK filings with the Securities and Exch ange Commission.

20. Ta ble P-20 contains quart erly sales ($MM) of The Gap for fiscal years 1980-2004. Plot Tue Gap sales data as a time series and examine its properties. The objec-

tive is to generate forecasts of sales f~r the fo~r ~uarters of 2?05. Select an appro- priate sm oothing m ethod for forecastmg and Justify your choice.

THE SOLAR ALTERNATIVE COMPANY4

The Solar Alternative Compan y is a b o ut to e nter its third year of operatio n. B ob and M ar y Johnson , '-ho both teach science in the lo ca l high school,

fo unded th e compan y. The Johnsons started the Solar Alternative Company to supplement their teaching income. Based on their research into solar

-i't:scase was contributed by William P. Darrow of Towson State University, Towson, Maryland.

146 CHAPTER 4 Moving Averages and Smoothing Methods

energy systems, they we re able to put toge ther a solar system for heating domestic hot water. The sys- tem consists of a 100-gallon fiberglass storage tank, two 36-foot solar pane ls, electronic controls, PVC pipe, and miscellaneous fittings.

The payback period on the system is 10 years. Although this situation does not present an attrac- tive investment opportunity from a financial point of view, there is s ufficient inte rest in the novelty of the concept to provide a moderate level of sales. The Johnsons clear about $75 on the $2,000 price of an installed system, after costs and expenses. Material and equipment costs account for 75% of the installed system cost. An advantage that helps to offset the low profit margin is the fact that the product is not profitable e nough to generate any significant competition from h eating contractors. The Johnsons operate the business out of their home. They have an office in the baseme nt, and their one-car garage is used exclusively to store the system components and mate rials. A s a result, over- head is at a minimum. The Johnsons enjoy a modest supple mental income from the company's opera- tion. The business also provides a number of tax advantages.

Bob and Mary are pleased with the growth of the business. Although sales vary from month to month , overall the second year was much better

TABLE 4 - 10

Month 2005 2006

January 5 17 February 6 14 March 10 20 April 13 23 May 18 30 June 15 38

ASSIGNMENT

1. Identify the model Bob a nd Mary should use as the basis for their business pla nning in 2007, and discuss why you selected this model.

than the first . Many of the second-year customers are neighbors of people who had purchased the sys- tem in the first year. Apparently, after seeing the system operate successfully for a year, others were willing to try the solar concept. Sales occur through- out the year. Demand for the system is greatest in late summer and early fall, when homeowners typi- cally make plans to winterize their homes for the upcoming heating season.

With the anticipated growth in the business, the Johnson s felt t hat they needed a sales forecast to manage effective ly in the coming year. It usually takes 60 to 90 da ys to receive storage tanks after placing the order. The solar pane ls are available off the shelf most of the year. H owever, in the late summer and throughout the fall , the lead time can be as much as 90 to 100 days. Although there is limited competition, lost sales are nevertheless a real possibility if the potential customer is asked to wait several months for installation. Perhaps more important is t he need to m ake accurate sales pro- jections to take a d va ntage of quantity discount buying. These factors, when combined with the high cost of system components and th e limited storage space ava ilable in the garage, make it nec- essary to deve lop a reliable forecast. The sales his- tory for the company's first two years is given in Table 4-10.

Month 2005 2006

July 23 44 August 26 41 September 21 33 October 15 23 November 12 26 December 14 17

2. Forecast sales for 2007.

CHAPTER 4 Moving Averages and Smoothing Methods 14 7

CASE 4-2 MR. TUX

John Mosby, owne r of several Mr. Tux rental stor es, is beginning to forecast his most important business l'ariable. monthly do llar sales (see the Mr. Tux cases in previous chapters). One of his emp loyees, Vtrginia Perot, gath e re d the sa les data shown in Case 2-2.John now wants to create a forecast based on these sales data using moving average and exponential smoothing techniques.

John used Minitab in Case 3-2 to determine !hat these data are both trending and seasonal. He has been told that simple m ovin g aver ages and exponential smoothin g techniques will not work ~ith these data but decides to find out for himself.

He begins by trying a three-month moving aver- age. The program calculates several summary fore- cast error measurements. These values s ummarize Ille errors found in predicting actu a l historical data 1alues using a three-month moving average. John decides to record two of these e rror measurements:

MAD= 54,373

MAPE = 47.0% The MAD (mean absolute deviation) is the average absolute error made in forecasting past values. Each forecast using the three- month moving average method is off by an average of 54,373. The MAP E (mean absolute percentage e rror) shows the error as a percentage of the actual value to be forecast. The average error using the three-month moving ave r- age technique is 47% , or almost half as large as the ralue to be forecast.

Next, John tries simple exponential smoothing. The program asks him to input the smoothing l1lnstant (a) to be used or to ask that the optimum a 1alue be ca lculated. Jo hn does the latter, and

OUESTIONS I. Summarize the forecast error level for the beSt

method John has found using Minitab. 2. John used the Minitab default values for a, /3,

and 'I- John thinks there are other c hoices for these parameters that would lead to smaller error measurements. Do you agree?

3· A~though disappointed with his initia l resu!ts, thtS may be the best h e can do with smoothrn g

the program finds the optimum a value to be .867. Again he records the appropria te error measurements:

MAD = 46,562

MAP£ = 44.0%

John asks the program to use Holt's linear exponential smoothing o n his data. This program uses the expo- nential smoothing method but can account for a trend in the data as well. John has the program use a smoothing constant of .4 for both a a nd {3. The two summary error measurements for Holt's method are

MAD= 63,579

MAPE = 59.0%

John is surprised to find larger measurement errors for this technique. He decides that the seasonal aspect of the data is the problem. Winters' multiplica- tive exponentia l smoothing is the next method John tries. This method can account for seasonal factors as well as trend. John uses smoothing constants of a = .2, {3 = .2, and 1' = .2. Error measurements are

MAD = 25,825

MAPE = 22.0 % When John sits down and begins studying the results of hi s analysis, he is disappointed. The Win ters' method is a big improvement; however, the MAP£ is still 22 % . He had hoped that one of the methods he used would result in accurate forecasts of past periods; he could the n use this method to forecast t he sales levels fo r coming months over the next year. But the average absolute errors (MADs) and percentage errors (MAPEs) for these methods lead him to look for anothe r way of forecasting.

methods. What should John do, for example, to determine the adequacy of the Winters ' fore- casting technique?

4. Although not calculated directly in Minitab, the MPE (mean percentage error) measures fore- cast bias. What is the ideal value for the MPE? What is th e implication of a negative sign on the MPE?

148 CH APTE R 4 Moving Averages and Smoothing Methods

CASE 4-3 CONSUMER CREDIT COUNSELING

The Consumer Credit Counseling (CCC) operation was described in Case 1-2. The executive director, Marv Harn ishfeger, concluded that the most impor- tant va riable that CCC needed to forecast was the numbe r of new clients that would be seen in the rest of 1993. Marv provided Dorothy Mercer monthly

ASSIGNMENT 1. D evelop a naive model to forecast the number

of new clients seen by CCC for the rest of 1993. 2. Develop a moving average model to forecast

the number of new clients see n by CCC for the rest of 1993.

3. D evelop an exponential smoothing procedure to forecast the number of new clients seen by CCC for the rest of 1993.

data for the number of new clients seen by CCC for the period from January 1985 th rough March 1993 (see Case 3-3). Dorothy then used autocorrelation analysis to explore the data pattern. Use the results of this investigation to complete the following tasks.

4. Evaluate these forecast ing methods using the forecast error summary measures presented in Chapter 3.

5. Choose the best model and forecast new clients for the rest of 1993.

6. Determine the adequacy of the forecasting model you have chosen.

CASE 4-4 MURPHY BROTHERS FURNITURE

Julie Murphy knows that most important operat- in g decisions depend, to some extent, o n a fore- cast. For M urphy Brothe rs F urniture, sales fore- casts impact adding new furniture lines or d ro pping o ld o nes; pla nnin g purch ases; setting sa les quotas; a nd making person nel, advertising, and fi nancia l decisions. Specifically, Julie is aware of several current forecasting needs. Sh e knows that the production department has to sched u le employees and de termine raw m aterial orders for the next month or two. She also knows that her dad , Glen Murp h y, needs to determine the best

QUESTIONS 1. D o any of the forecasting models studied in this

chapter work with the national sales data? 2. D o any of the forecasting models studied in this

chapter work with the actual Murphy Brothers' sales data?

in vestment opportunities and must forecast the demand for a new furniture line.

In Case 3-lA, Julie Murphy used monthly sales for all retail stores from 1983 through 1995 (see Table 3-8} to develop a pattern for Murphy Brothers Furniture sales. In Case 3-lB, Glen Murphy discovered actual sales data for the past four years, 1992 through 1995 (see Table 3-9). Julie was not excited about her father"s discovery because she was not sure which set of data to use to develop a forecast for 1996. She detennined that sales for all retail stores had somewhat the same pat- tern as actual Murphy Brothers' sales data.

3. Which data set and forecasting model should Julie use to forecast sales for 1996?

CHAPTER 4 Moving Averages and Smoothing Methods 149

CASE 4-5 FIVE-YEAR REVENUE PROJECTION FOR DOWNTOWN RADIOLOGY

Some years ago Downtown Radiology developed a medical imaging center that was more complete and technologically advanced than any located in an area of eastern Washington and northern Idaho called the Inland Empire. The equipment planned for the center equaled or surpassed the imaging facilities of all med- ical centers in the region. The center initially con- tained a 9800 series CT scanner and nu clear magnetic resonance imaging (MRI) equipment. The center also mcluded ultrasound, nuclear medicine, digital sub- traction angiography (DSA), mammography, and conventional radiology and fluoroscopy equipment. Ownership interest was made available in a type of public offering, and Downtown Radiology used an mdependent evaluation of the market. Professional ~larketing Associates, Inc., evaluated the market and completed a five-year projection of revenue.

STATEM ENT OF THE PROBLEM The purpose of this study is to forecast revenu e for the next five years for the proposed medical imaging center, assuming you are employed by Professional ~farketing Associates, Inc., in the year 1984.

OBJECTIVES The objectives of this study are to

• Identify market areas for each type of medical procedure to be offered by th e new facility.

' Gather and analyze existing data on market area revenue for each type of procedure to be offered by the new facility.

' Identify trends in the health care industry that will positively or negatively affect revenue of proce- dures to be provided by the proposed facility.

• Identify factors in the business, mar keting, or facility planning of the new venture that will pos- itively or negatively impact revenue projections.

• Analyze past procedures of D owntown Radiology whe n compiling a database for the forecasting model to be developed.

• Utilize the appropriate qu antitative for~cas_ting model to arrive at five-year revenue pro1ect1ons for the proposed center.

M ETHO DOLOGY Medical Procedures The following steps were imple mented in order to complete the five-year projection of revenue. An analysis of the past number of procedures was per- formed . The appropriate forecasting model was developed and used to determine a starting point for the projection of each procedure.

1. The market area was determined for each type of procedure, and population forecasts were obtained for 1986 and 1990.

2. Doctor referral patterns were studied to deter- mine the percentage of doctors who refer to Downtown Radiology and the average number of referrals per doctor.

3. National rates were acquired from the National Center for Health Statistics. These rates were compared with actual numbers obtained from the Hospital Commission.

4. Downtown Radiology's market share was calcu- lated based o n actual CT scans in t he market area. (Market share for other procedures was determined based on Downtown R adiology's share compared to rates provided by the National Center for Health Statistics.)

Assumptions Toe following assumptions, which were necessary to develop a quantitative forecast, were made:

• The new imaging center will be operational, with all equipment functiona l except the MRI, on January 1, 1985.

• The nuclear magnetic resonance imaging equip- ment will be functional in April 1985.

• The offering of the limited partnership will be s uccessfully marketed to at least 50 physicians in the service area.

• Physicians who have a financial interest in the new imaging center will increase their referrals to the center.

• There will be no other MRis in the market area before 1987.

• The new imaging center will offer services at lower prices than the competition.

1 48 CHAPTER 4 Moving Averages and Smoothing Methods

CASE 4-3 CONSUMER CREDIT COUNSELING

The Consumer Credit Counseling (CCC) operation was described in Case 1-2. The executive director, Marv Harnishfeger, concluded that the most impor- tant variable that CCC needed to forecast was the number of new clients that would be seen in the rest of 1993. Marv provided Dorothy Mercer monthly

ASSIGNMENT l. Develop a naive model to forecast the number

of new clients seen by CCC for the rest of 1993. 2. Develop a moving average model to forecast

the number of new clients seen by CCC for the rest of 1993.

3. Develop an exponential smoothing procedure to forecast the number of new clients seen by CCC for the rest of 1993.

data for the number of new clients seen by CCC for the period from January 1985 through March 1993 (see Case 3-3). Dorothy then used autocorrelation analysis to explore the data pattern. Use the results of this investigation to complete the following tasks.

4. Evaluate these forecasting methods using the forecast error summary measures presented in Chapter 3.

5. Choose the best model and forecast new clients for the rest of 1993.

6. Determine the adequacy of the forecasting model you have chosen.

CASE 4-4 MURPHY BROTHERS FURNITURE

Ju lie Murphy knows that most important operat- ing decisions depend, to some extent, on a fore- cast. For Murphy Brothers Furniture, sales fore - casts impact adding new furniture lines or dropping old ones; planning purchases ; setting sales quotas; and making personnel, advertising, and financial decisions. Specifically, Julie is aware of several current forecasting needs. She knows that the production department has to schedule employees and determine raw material orders for the next month or two. She also knows that her dad, Glen Murphy, needs to determine the best

QUESTIONS l. Do any of the forecasting models studied in this

chapter work with the national sales data? 2. Do any of the forecasting models studied in this

chapter work with the actual Murphy Brothers' sales data?

investment opportunities and must forecast the demand for a new furniture line.

In Case 3-lA, Julie Murphy used monthly sales for all retail stores from 1983 through 1995 (see Table 3-8) to develop a pattern for Murphy Brothers Furniture sales. In Case 3-lB, Glen Murphy discovered actual sales data for the past four years, 1992 through 1995 (see Table 3-9). Julie was not excited about her father's discovery because she was not sure which set of data to use to develop a forecast for 1996. She determined that sales for all retail stores had somewhat the same pat· tern as actual Murphy Brothers' sales data.

3. Which data set and forecasting model should Julie use to forecast sales for 1996?

CHAPTER 4 Moving Averages and Smoothing Methods 1 49

CASE 4-5 FIVE-YEAR REVENU E PROJECTION FOR DOWNTOWN RADIO LOGY

Some years ago Downtown Radiology developed a medical imaging center that was more complete and technologically advanced than any located in an area of eastern Washington and northern Idaho called the Inland Empire. The equipment planned for the center equaled or surpassed the imaging facilities of all med- ical centers in the region. The center initially con- tained a 9800 series CT scanner and nuclear magnetic resonance imaging (MRI) equipment. The center also included ultrasound, nuclear medicine, digital sub- traction angiography (DSA), mammography, and conventional radiology and fluoroscopy equipment. Ownership interest was made available in a type of public offering, and Downtown Radiology used an independent evaluation of the market. Professional Marketing Associates, Inc., evaluated the market and completed a five-year projection of revenue.

STATEMENT OF THE PROBLEM The purpose of this study is to forecast revenue for the next five years for the proposed medical imaging center, assuming you are employed by Professional Marketing Associates, Inc., in the year 1984.

OBJECTIVES The objectives of this study are to

• Identify market areas for each type of medical procedure to be offered by the new facility.

• Gather and analyze existing data on market area revenue for each type of procedure to be offered by the new facility.

• Identify trends in the health care industry that will positively or negatively affect revenue of proce- dures to be provided by the proposed facility.

• Identify factors in the business, marketing, or facility planning of the new venture that will pos- itively or negatively impact revenue projections.

• Analyze past procedures of Downtown Radiology when compiling a database for the forecasting model to be developed .

• Utilize the appropriate quantitative forecasting model to arrive at five-year revenue projections for the proposed center.

METHODOLOGY Medical Procedures The following steps were implemented in order to complete the five-year projection of revenue. An analysis of the past number of procedures was per- formed. The appropriate forecasting model was developed and used to determine a starting point for the projection of each procedure.

1. The market area was determined for each type of procedure, and population forecasts were obtained for 1986 and 1990.

2. Doctor referral patterns were studied to deter- mine the percentage of doctors who refer to Downtown Radiology and the average number of referrals per doctor.

3. National rates were acquired from the National Center for Health Statistics. These rates were compared with actual numbers obtained from the Hospital Commission.

4. Downtown Radiology's market share was calcu- lated based on actual CT scans in the market area. (Market share for other procedures was determined based on Downtown Radiology's share compared to rates provided by the National Center for Health Statistics.)

Assumptions The following assumptions, which were necessary to develop a quantitative forecast, were made:

• The new imaging center will be operational, with all equipment functional except the MRI, on January 1, 1985.

• The nuclear magnetic resonance imaging equip- ment will be functional in April 1985.

• The offering of the limited partnership will be successfully marketed to at least 50 physicians in the service area.

• Physicians who have a financial interest in the new imaging center will increase their referrals to the center.

• There will be no other MRis in the market area before 1987.

• The new imaging center will offer services at lower prices than the competition.

1 50 CHAPTER 4 Moving Averages and Smoothing Methods

• An effective marketing effort will take place, especially concentrating on large employers, insurance groups, and unions.

• The MRI will replace approximately 60% of the head scans that are presently done with the CT scanner during the first six months of operation and 70% during the next 12 months.

• The general public will continue to pressure the health care industry to hold down costs.

• Costs of outlays in the health care industry rose 13.2% annually from 1971 to 1981. The Health Care Financing Administration estimates that the average annual rate of increase will be reduced to approximately 11 % to 12% between 1981 and 1990 (Industry Surveys, April 1984).

• Insurance firms will reimburse patients for (at worst) 0% up to (at best) 100% of the cost of magnetic resonance imaging (Imaging News, February 1984).

Models A forecast was developed for each procedure, based on past experience, industry rates, and rea- sonable assumptio ns. The models were developed based on the preceding assumptions; however, if the assumptions are not valid, the models will not be accurate.

A NALYSI S O F PA ST DATA Office X-Rays The number of X-ray procedures performed was analyzed from July 1981 to May 1984. The data included diagnostic X-rays, gastrointestinal X-rays, breast imaging, injections, and special procedures. Examination of these data indicated that no trend or seasonal or cyclical pattern is present. For this rea- son, simple exponential smoothing was chosen as the appropriate forecasting method. Various smoothing constants were examined, and a smoothing constant of .3 was found to provide the best model. The results are presented in Figure 4-14. The forecast for period 36, June 1984, is 855 X-ray procedures.

Office Ultrasound The number of ultrasound procedures performed was analyzed from July 1981 to May 1984. Figure4-15 shows the d a ta pattern. Again, no trend or seasonal or cyclical pattern was present. Exponential smooth- ing with a smoothing constant of a = .5 was deter- mined to provide the best model. The forecast for June 1984 plotted in Figure 4-15 is 127 ultrasound procedures.

The number of ultrasound procedures performed by the two mobile units owne d by Downtown Radiology was analyzed from July 1981 to May 1984. Figure 4-16 s hows the data pattern. An increasing

FIGURE 4- 14 Simple Exponential Smoothing: Downtown Radiology X-Rays

Simple Exponential Smoothing for Office X- Rays

1750 1/oriable

- Actual

--- FRs - +- Forecasts

1500 --.I. - '35-0% PI

"' > !II

1250 a: I

X .... ... (IJ u

t;: lCXXl -0

Smoothin,;j Const•nt Alph• 0 .3

AcclW KY Me.astXu MAPE 14.2 MAO 133.7 MSO 33765.7

750

500 ... 4 8 12 16 20 24 28 32

Time 36

160

150

140

130 "0 C:

120 ::, 0

"' (0 110 ... ..,

3 100

90

80

70

CHAPTER 4 Moving Averages and Smoothing Methods 1 5 1

S imple Exponential Smoothing for Office Ultr asound

4 8 12

1/ariable -+- Actu.1 -- Fib -+- Forecosts ......A. - 95.0% PI

I Smoothing Const.nt I ♦ . Alpha 0.5 .

16 20 24 28 32 36 Time

Ac cur K Y Meuures MAPE 12.211 MAO 12.035 MSD 242.739

FIGURE 4-15 Simple Exponential Smoothing: Downtown Radiology Ultrasound

Holt's Linear Smoothing for Nonoffice Ultrasound

~ ::,

I 5 Cl) u

t;: -~ 0 z

300 •

250

•

200

•

V...;.blo

-+- Actuol

---- Fb - + - Forecists ...,._ 'lS .O'll> Pl

Smoottw.J Constonts Alph,(lml) 05 G.,,,,... (trend) 0. 1

AcC\wacy Ml:.-st.rH MAPE 14.45 MAD 25.85

MSO 1064,17

1001.__-,---.--..---,---,r---,--.--.----,-J 12 16 20 24 28 32 36 4 8

Time

FIGURE 4-16 Holt's Linear Exponential Smoothing: Downtown Radiology Nonoffice Ultrasound

trend is apparent and can be modeled using Holt's ~ a-parameter linear exponential smoothing. Smooth- mg constants of a = .5 and f3 = .1 are used, and the forecast for period 36, June 1984, is 227.

Nuclear Medicine Procedures Toe number of nuclear medicine procedures perfo rmed by the two mobile units owned by

Downtown Radiology was analyzed from August 1982 to May 1984. Figure 4-17 shows the data pat- tern. The data were not seasonal and had no trend or cyclical pattern. For this reason, simple exponential smoothing was chosen as the appropriate forecast- ing method. A smoothing factor of a = .5 was found to provide the best model. The forecast for period 23, June 1984, is 48 nuclear medicine procedures.

1 52 CHAPTER 4 Moving Averages and Smoothing Methods

Office CT Scans

Simple Exponential Smoothing for Nonoffice Nuclear Medicine V•riable

---Actuol 160 -- Fils

- -+-- Forecasts Q)

_. - 95.0% Pl

C: I Smoothing Const.nt I ·o 120 a '5 Q) \

Alpha 05

~ I\ Accuracy Measures .. \ t'0 MAPE s,.02 ~ 80

.__ 25.48

'II MAO :,

\ MSO 1021 .0, z Cl)

._ ♦ u

i::: 40 ... 0 C: 0 z

0

2 4 6 8 10 12 14 16 18 20 22 Time

FIGURE 4-17 Simple Exponential Smoothing: Downtown Radiology Nonoffice Nuclear Medicine

The number of CT scans performed was also analyzed from July 1981 to May 1984. Seasonality was not found, and the number of CT scans did not seem to have a trend. However, a cyclical pattern seemed to be present. Knowing how many scans were performed last month would be important in forecasting what is going to happen this month. An autoregressive model

(see Chapters 8 and 9) was examined and compared to an exponential smoothing model with a smoothing constant of a = .461. The larger smoothing constant gives the most recent observation more weight in the forecast. The exponential smoothing model was deter- mined to be better than the autoregressive model, and Figure 4-18 shows the projection of the number of CT scans for period 36, June 1984, to be 221.

Simple Exponential Smoothing for CT Scans

350 Variable -Actual

--- Fits - ♦- Forecasts

300 " • " \

A

<LI l,

C: ~ (0

250 ',, u er., \ I- c.J ..

---A - 95 ,0% PI

Smoo thing Constant Alpha 0.4&0539

Accuracy Meas1Xe.s MAPE 113 5 MAD 27 .43 MSD 1208.42

\ ♦

200

150 4.

4 8 12 16 20 24 28 32 36 Time

FIGURE 4-18 Simple Exponential Smoothing: Downtow R d- I CT Scans n a 10 ogy

CHAPTER 4 Moving Averages and Smoothing Methods 1 53

MARKET AREA ANALYSIS Market areas were determined for procedures cur- rently done by Downtown Radiology by examining patient records and doctor referral patterns. Market areas were determined for procedures not currently done by Downtown Radiology by investigating the competition and analyzing the geographical areas they served.

CT Scanner Market Area The market area for CT scanning for the proposed medical imaging center includes Spokane, Whitman, Adams, Lincoln, Stevens, and Pend Oreille Counties in Washington and Bonner, Boundary, Kootenai, Benewah, and Shoshone Counties in Idaho. Based on the appropriate percentage projections, the CT scanning market area will have a population of 630,655 in 1985 and 696,018 in 1990.

Quantitative Estimates To project revenue, it is necessary to determine certain quantitative estimates. The most important estimate involves the number of doctors who will par- ticipate in the limited partnership. The estimate used in future computations is that at least 8% of the doc- tor population of Spokane County will participate.

The next uncertainty that must be quantified involves the determination of how the referral pat- tern will be affected by the participation of 50 doc- tors in the limited partnership. It is assumed that 30 of the doctors who presently refer to Downtown Radiology will join the limited partnership. Of the 30 who join, it is assumed that 10 will not increase their referrals and 20 will double their referrals. It is also assumed that 20 doctors who had never referred to Downtown Radiology will join the lim- ited partnership and will begin to refer at least half of their work to Down town Radiology.

The quantification of additional doctor referrals should be clarified with some qualitative observa- tions. The estimate of 50 doctors joining the pro- posed limited partnership is conservative. There is a strong possibility that doctors from areas outside of Spokane County may join. Traditionally, the doctor referral pattern changes very slowly. However, t~e sudden competitive nature of the marketplace will probably have an impact on doctor referrals. If the li~ited partnership is marketed to doctors ~n spe- cialties with high radiology referral potential, the number of referrals should increase more than projected. The variability in the number of doctor

referrals per procedure is extremely large. A few doctors referred an extremely large percentage of the procedures done by Downtown Radiology. If a few new many-referral doctors are recruited, they can have a major effect on the total number of pro- cedures done by Downtown Radiology.

Finally, the effect that a new imaging center will have on Downtown Radiology's market share must be estimated. The new imaging center will have the best equipment and will be prepared to do the total spectrum of procedures at a lower cost. The number of new doctors referring should increase on the basis of word of mouth from the new investing doctors. If insurance companies, large employers, and/or unions enter into agreements with the new imaging center, Downtown Radiology should be able to increase its share of the market by at least 4% in 1985, 2% in 1986, and 1 % in 1987 and retain this market share in 1988 and 1989. This market share increase will be referred to as the total imaging effect in the rest of this report.

Revenue Projections Revenue projections were completed for every pro- cedure. Only the projections for the CT scanner are shown in this case.

CT Scan Projections Based on the exponential smoothing model and what has already taken place in the first five months of 1984, the forecast of CT scans for 1984 (January 1984 to January 1985) is 2 ,600.

The National Center for Health Statistics reports a rate of 261 CT scans per 100,000 population per month . Using the population of 630,655 projected for the CT scan market area, the market should be 19,752 procedures for all of 1985. The actual number of CT scans performed in the market area during 1983 was estimated to be 21,600. This estimate was based on actual known procedures for Downtown Radiology (2,260), Sacred Heart (4,970), Deaconess (3,850), Valley (2,300), and Kootenai (1,820) and on estimates for Radiation Therapy (2,400) and Northwest Imaging (4,000). If the estimates are accurate, Downtown Radiology had a market share of approximately 10.5% in 1983. The actual values were also analyzed for 1982, and Downtown Radiology was projected to have approximately 15.5% of the CT scan market during that year. Therefore, Downtown Radiology is forecast to aver- age about 13% of the market.

154 CHAPTER 4 Moving Averages and Smoothing Methods

Based on the increased referrals from doctors belonging to the limited partnership and an analysis of the average number of referrals of CT scans, an increase of 320 CT scans is projected for 1985 from this source. If actual values for 1983 are used, the rate for the Inland Empire CT scan market area is 3,568 (21 ,600/6.054) per 100,000 population. If this pattern continues, the number of CT scans in the market area will increase to 22,514 (3,568 X 6.31) in 1985. Therefore, Downtown Radiology's market share is projected to be 13% (2,920/22,514). When the 4 % increase in market share based on total imaging is added, Downtown Radiology's market share increases to 17.0% , and its projected number of CT scans is 3,827 (22,514 X .17).

However, research seems to indicate that the MRI will eventually replace a large number of CT head scans (Applied Radiology, May/June 1983, and Diagnostic Imaging, February 1984). The National Center for Health Statistics indicated that 60% of all CT scans were of the head. Downtown Radiology records showed that 59% of its CT scans in 1982 were head scans and 54% in 1983. If 60% of Downtown Radiology's CT scans are of the head and the MRI approach replaces approximately 60% of them, new projections for CT scans in 1985 are necessary. Since the MRI will operate for only half the year, a reduc- tion of 689 (3,827 / 2 X .60 X .60) CT scans is forecast.

The projected number of CT scans for 1985 is 3,138. The average cost of a CT scan is $360, and the projected revenue from CT scans is $1,129,680. Table 4-11 shows the projected revenue from CT scans

QUESTION 1. Downtown Radiology's accountant projected

that revenue would be considerably higher than that provided by Professional Marketing Associates. Since ownership interest will be made available in some type of public offering,

TABLE 4-11 Five-Year Projected Revenue for CT Scans

Year Procedures Revenue ($)

1985 3,138 1,129,680

1986 2 ,531 1,012,400

1987 2 ,716 1,205,904

1988 2,482 1,223,626

1989 2 ,529 1,383,363

for the next five years. The cost of these procedures is estimated to increase approximately 11 % per year.

Without the effect of the MRI, the projection for CT scans in 1986 is estimated to be 4,363 (6.31 x 1.02 X 3 ,568 X .19) . However, if 60% are CT head scans and the MRI replaces 70% of the head scans, the projected number of CT scans should drop to 2 ,531 [4,363 - ( 4 ,363 x .60 x .70)].

The projection of CT scans without the MRI effect for 1987 is 4 ,683 (6.31 x 1.04 X 3,568 x .20). The forecast with the MRI effect is 2,716 [4,683 - ( 4 ,683 X .60 X .70) ).

The projection of CT scans without the MRI effect for 1988 is 4 ,773 (6.31 x 1.06 X 3,568 x .20). The forecast with the MRI effect is 2,482 [4,773 - (4,773 X .60 X .80) ).

The projection of CT scans without the MRI effect for 1989 is 4 ,863 (6.31 X 1.08 X 3 ,568 X .20). The forecast with the MRI effect is 2,529 [4.863 - ( 4,863 X .60 X .80) ).

Downtown Radiology 's management roust make a decision concerning the accuracy of Professional Marketing Associates' projections. You are asked to analyze the report. What rec· ommendations would you make?

CASE 4-6 WEB RETAILER

Example 1.2 introduced Pat Niebuhr and his team, who are responsible for developing a global staffing plan for the contact centers of a large web retailer. Pat

needs to take a monthly forecast of the total orders and contacts per order (CPO) supplied by the finance department and ultimately forecast the number of

CHAPTER 4 Mo ving A verages and Smoothing Methods 1 55

customer contacts (phone, email, and so forth) arriv- ing at the retailer's contact centers weekly. The contact centers are open 24 hours 7 days a week and must be appropriately staffed to maintain a high service level. The retailer recognizes that excellent customer service will likely result in repeat visits and purchases.

Pat thinks it might be a good idea to use the historical data to generate forecasts of orders and contacts per order directly. He is interested in determining whether these forecasts are more accu- rate than the forecasts for these quantities that the finance department derives from revenue projec- tions. A s a start, Pat and his team are interested in the data patterns of the monthly historical orders and contacts per order and decide to plot these time series and analyze the autocorrelations. The data are given in Table 4-12 and plotted in Figures 4-19 and 4-20. The autocorrelations are shown in Figures 4-21 and 4-22.

The key equation for Pat and his team is

Contacts = Orders X CPO

Historical data provide the percentage of contacts for each day of the week. For example, historically 9.10% of the weekly contacts occur on Sundays, 17.25% of the weekly contacts occur on Mondays, and so forth. Keeping in mind the number of Sundays, Mondays, and so on in a given month, monthly forecasts of contacts can be converted t o weekly forecasts of contacts. It is the weekly fore- casts that are used for staff planning purposes.

P a t is intrigued with the time series plots and the autocorrelation functions and feels a smoothing procedure might be the right tool for fitting the time series for orders and contacts per order and for gen- erating forecasts.

TABLE 4-1 2 Orders and Contacts per Order (CPO) for Web Retailer, June 2001-June 2003

Month Orders CPO

Jun-01 3,155,413 0.178

Jul-01 3,074 ,723 0.184

Aug-01 3,283,838 0.146

Sep-01 2,772,971 0.144

Oct-01 3,354,889 0.144

Nov-01 4,475,792 0.152

Dec-01 5,944,348 0.152

Jan-02 3,742 ,334 0.174

Feb-02 3,681,370 0.123

Mar-02 3,546,647 0.121

Apr-02 3,324,321 0.117

May-02 3,318,181 0.116

Jun-02 3,181,115 0.129

Jul-02 3,022,091 0.131

Aug-02 3,408,870 0.137

Sep-02 3,501 ,779 0.140

Oct-02 3,712,424 0.144

Nov-02 4,852,090 0.129

Dec-02 7,584,065 0.124

Jan-03 4,622,233 0.136

Feb-03 3,965,540 0.116

Mar-03 3,899,108 0.111

Apr-03 3,670,589 0.108

May-03 3,809,110 0.101

Jun-03 4,159,358 0.105

1 56 CHAPTER 4 Moving Averages and Smoothing Methods

QUESTIONS

"' ... Q)

"C ... 0

8000000

7000000

6000000

5000000

4COJOOO

300CO)()

Month Jun

Year 2001

Time Series Plot of Orders

Oct Feb 2002

Jun Oct Feb 2003

Jun

FIGURE 4-19 Time Series Plot of Orders, June 2001-June 2003

0.19

0.18

0.17

0.16

0.15 0 C. t.l 0.14

0.13

0.12

0.11

0.10

Month

Year

Time Series Plot of CPO

Jun oct Feb Jun 2001 2002

Oct Feb 2003

.l.Jrl

FIGURE 4 -20 Time Series Plot of Contacts per Order (CPO) J 2001-June 2003 ' une

1. What did Pat and his team learn about the data patterns for orders and contacts per order from the time series plots and autocorrelation functions?

2. :it an a~propriate smoothing procedure to the rders time series and generate forecasts for

the next four months. Justify your choice.

Autocorrelation Function: Orders

Lag ACF T LBQ l 0.449007 2.25 5.67 2 0.052358 0.22 5 . 75 3 - 0 .101155 -0 . 43 6 . 06 4 -0 . 196536 -0 . 82 7.31 5 -o. 271190 -1.10 9 . 79 6 -0.187990 -0.73 11. 04

CHAPTER 4 Moving Averages and Smoothing Methods 1 5 7

• I I '

Autocorrelation Function for Orders 1,0 ,-------------------,

0.8

0.o

g 0.4 'i; 0.2

__________________ .,.._ f O.Ot..__ __ _,_ __ --,---,-----r----.--l 8 0 -0.2 $ < -0,4

-0.o

-0.8

-1.0

FIGURE 4 -21 Autocorrelation Function for Orders

- -- - - - - --- - Autocorrelation for CPO , ,.,,J? •• :~~,.;;;:---

Autocorrelation Function: CPO

Lag ACF T LBQ l 0.637679 3 .19 11.44 2 0.346901 1.29 14. 97

3 0.195365 0 .68 16 . 14

4 0.098064 0.34 16.45

1.0

0.8

o.&

Autocorrelation Function for CPO

------ ------· 5 0. 084642 0 . 29 16.69

6 0.147248 a.so 17 . 46

0,4 i---

1

l 0.2 I I = O,Ot-'----__._ __ __,_ ___ .J.I ___ L_l ___ ~I e .e -0.2 "

I C -0.4 --- - --- -0.& -- - -0.8

-1,0

2 4 Log

FIGURE 4-22 Autocorrelation Function for Contacts per Order (CPO)

3- Fit an appropriate smoothing procedure to the contacts per order time series and generate fore- casts for the next four months. Justify your choice.

4. Use the results for Questions 2 and 3 to generate forecasts of contacts for the next four months.

S. Pat has access to a spreadsheet with historical actual con tacts. He is considering forecasting

6.

contacts directly instead of multiplying forecasts of orders and contacts per order to get a fore- cast. Does this seem reasonable? Why? Many orders consist of more than one item (unit). Would it be better to focus on number of units and contacts per unit to get a forecast of contacts? Discuss.

1 58 CHAPTER 4 Moving Averages and Smoothing Methods

CASE 4-7 SOUTHWEST MEDICAL CENTER

Mary Beasley is responsible for keeping track of the number of billable visits to the Medical Oncology group at Southwest Medical Center. Her anecdotal evidence suggests that the number of visits has been increasing and some parts of the year seem to be busier than others. Some doctors are beginning to complain about the work load and suggest they don't always have enough time to interact with indi- vidual patients. Will additional medical staff be required to handle the apparently increasing demand?

If so, how many new doctors should be hired and/or be reassigned from other areas?

To provide some insight into the nature of the demand for service, Mary opens her Excel spreadsheet and examines the total number of billable visits on a monthly basis for the last several fiscal years. The data are listed in Table 4-13.

A time series plot of Mary's data is shown in Figure 4-23. As expected, the time series shows an upward trend, but Mary is not sure if there is a

TABLE 4·13 Total Billable Visits to Medical Oncology, FYI 995-FY2004

Year Sept.

FY1994-95 725 FY1995-96 899 FY1996-97 916 FY1997-98 1,061 FY1998-99 1,554 FY1999-00 1,492 FY2000-01 1,018 FY2001-02 1,083 FY2002-03 1,259 FY2003-04 1,339

., .., :sa > <6 .., 0 ...

Oct. Nov. Dec. Jan. Feb. Mar. Apr. May

789 893 823 917 811 1,048 970 1,082 1,022 895 828 1,011 868 991 970 934

988 921 865 998 963 992 1,118 1,041 1,049 829 1,029 1,120 1,084 1,307 1,458 1,295 1,472 1,326 1,231 1,251 1,092 1.429 1,399 1,341 1,650 1,454 1,373 1,466 1,477 1,466 1,182 1,208 1,233 1,112 1,107 1,305 1,181 1,391 1,324 1,259 1,404 1,329 1,107 1,313 1,156 1,184 1,404 1,310 1,295 1,100 1,097 1,357 1,256 1,350 1,318 1,271 1,351 1,197 1,333 1,339 1,307

Time Series Plot of Total Visits to Medical Oncology

1600 -tt --- + - 1400 ti ' f'li "'I --.,. _ r \. • t p. 'iw -+--

l ~ ' ,1+ t 1\) &~ ,,. _._ t I ~ 11I '1J \ ~) ,, Ji \,

.....,.._ 1200 ~· 1Pili ! -+-I •H t -----, 11,+ 1000 v\~\" .. ~ i v• i V

800 1 ..t.

600 7"--::-:---:::-------:~~--,---,r---,---.---,--..--_J 1 11 22 33 44 55 66 77 88 99 110

Month

FIGURE 4 ·23 Time Series Plot of Total Visits to Medical Oncology, FVl 995- FY2004

Jun. Jul. Aug.

1,028 1,098 1,062 784 1,028 956

1,057 1,200 1,062 1,412 1,553 1,480 1,409 1,367 1,483 1,132 1,094 1,061 1,236 1,227 1,294 1,200 1,396 1,373 1,439 1,441 1.352

FascarYr I 2 3 4 5 6 7 8 9

10

CHAPTER 4 Moving Averages and Smoothing Methods 1 59

seasonal component in the total visits series. She decides to investigate this issue by constructing the autocorrelation function. If a seasonal component exists with monthly data, Mary expects to see fairly large autocorrelations at the seasonal lags: 12, 24,

QUESTIONS l. What did Mary's autocorrelation analysis

show? 2. Fit an appropriate smoothing procedure to

Mary's data, examine the residual autocor- relations, and generate forecasts for the remain- der of FY2003-04. Do these forecasts seem reasonable?

and so forth. Mary knows from a course in her Executive MBA program that Winters' smoothing procedure might be a good way to generate fore- casts of future visits if trend and seasonal compo- nents are present.

3. Given the results in Question 2, do you think it is likely another forecasting method would gen- erate "better" forecasts? Discuss.

4. Do you think additional medical staff might be needed to handle future demand? Write a brief report summarizing Mary's data analysis and the implications for additional staff.

CASE 4-8 SURTIDO COOKIES

In Case 3-5, Jame Luna investigated the data pattern of monthly Surtido cookie sales (see Table 3-12). In that case, Karin, one of the members of Jame's team, suggested forecasts of future sa les for a given month might be generated by simply using the historical average sales for that month. After learning some- thing about smoothing methods however, Jame thinks a smoothing procedure might be a better way to construct forecasts of future sales. Jame recog-

QUESTIONS l. What pattern(s) did Jame observe from a time

series plot of Surtido cookie sales? 2. Are the autocorrelations consistent with the pat-

tern(s) Jame observed in the time series plot? 3. Select and justify an appropriate smoothing

procedure for forecasting future cookie sales

Minitab Applications

nizes that important first steps in selecting a fore- casting method are plotting the sales time series and conducting an autocorrelation analysis. He knows you can often learn a lot by simply examining a plot of your time series. Moreover, the autocorrelations tend to reinforce the pattern observed from the plot. Jame is ready to begin with the goal of generating forecasts of monthly cookie sales for the remaining months of 2003.

and produce forecasts for the remaining months of 2003.

4. Use Karin's historical monthly average sugges- tion to construct forecasts for the remaining months of 2003. Which forecasts, yours or Karin's, do you prefer? Why?

The problem. In Example 4.3, the Spokane Transit Authority data need to be forecast using a five-week moving average.

Minitab Solution 1. Enter the Spokane Transit Authority data shown in Table 4-2 (see p. 112) into

column Cl. Click on the following menus:

Stat>Time Series>Moving Average

1 60 CHAPTER 4 Moving Averages and Smoothing Methods

2. The Moving Average dialog box appears. . . . . a. Double-click on the variable Gallons and it will appear to the nght ofVanable. b. Since we want a five-month moving average, indicate 5 for MA length. c. Do not click on the Center moving average box. We will use a centered moving

average to smooth data in Chapter 5. . d. Click on Generate forecasts and indicate 1 to the nght of Number of forecasts. e. Click on OK and Figure 4-4 will appear.

The problem. In Example 4.6, the Acme Tool Company data need to be forecast using single exponential smoothing.

Minitab Solution 1. Enter the Acme Tool Company data shown in Table 4-1 (seep. 109) for the years

2000 through 2006 into column Cl. Click on the following menus:

Stat>Time Series>Single Exponential Smoothing

2. Toe Single Exponential Smoothing dialog box Appears. a. Double-click on the variable Saws and it will appear to the right of Variable. b. Under Weight to Use in Smoothing, choose Optimal ARIMA and then click on

OK. The result is shown in Figure 4-8 (see p . 129).

The problem. In Example 4.10, the Acme Tool Company data need to be forecast using exponential smoothing adjusted for trend and seasonality.

Minitab Solution l. Enter the Acme Tool Company data shown in Table 4-1 (see p.109) for the years

2000_ through 2006 in column Cl. Click on the following menus:

Stat>Time Series>Winters' Method

FIGURE 4-24 Minitab Winters' Method Dialog Box

Variable: Saws .=======:!I

Seasonal length: 14 -=-===;:;,!,!

Method Type

• Multiplicative Additive

~ Gener ate forecasts Number of forecasts:

starting from origin:

Time ...

Weights to Use in Smoothing

Level: 0.4

Trend: 0.1

Seasonal: 0,3

Options,,. Storage ...

Graphs ... Resuts ...

OK Cancel

2.

CHAPTER 4 Moving Averages and Smoothing Methods 1 61

The Winte~s' Meth_od dialog box appears, as shown in Figure 4-24. a. The vanable of mterest is Saws. b. Since th_e data are quarterly, indicate 4 for Seasonal length. c. The Weights to Use in Smoothing are Level: 0.4; Trend: 0.1; and Seasonal: 0.3. d. Click on Generate forecasts, and for Number of forecasts, indicate 4. e. Click on Storage.

3. The Winters' Method Storage dialog box appears. a. Click on Level estimates, Trend estimates, Seasonal estimates, Fits ( one-period-

ahead forecasts) , and R esiduals. b. Click on OK on both the Winters' Method Storage dialog box and the Winters'

Method dialog box. The results are shown in Table 4-9 and Figure 4-12. The forecast for the first quarter of 2007 is 778.2.

4. To store the data for further use, click on the following menus:

File>Save Worksheet As

5. The Save Worksheet As dialog box appears.

Excel Applications

a. Type a name such as Saws in the File Name space. b. The Save as Type space allows you to choose how you want to save your file.

Most of the time you will select Minitab. However, you can save your file so sev- eral software programs can use it. A s an example, you could choose to save the file as an Excel file. The file is saved as Saws.xis and will be used in the Excel Applications section .

The problem. In Example 4.5, the Acme Tool Company data were forecast using sin- gle exponential smoothing with a smoothing constant equal to .6.

Excel Solution 1. Open the file containing the data presented in Table 4-1 (see p. 109) by clicking on

the following menus:

2.

4.

File>Open

Look for the file called Saws.xls. Click on the following menus:

Tools>Data Analysis

Toe Data Analysis dialog box appears. Under Analysis Tools, choose Exponential Smoothing and click on OK. The Exponential Smoothing dialog box, shown in

Figure 4-25, appears. . a. Enter A2:A25 in the Input Range edit box. b. Check the Labels box. c. Enter .4 in the Damping factor edit box, since the damping factor (1 - a) is

defined as the complement of the smoothing constant. A d . Enter B3 in the Output Range edit box. (This will put the forecast Yr opposite

the corresponding value in column A) e. Check the Chart Output box. f. Now click on OK. The results (column B) and the graph are shown in Figure 4-26. Note that the Exponential Smoothing analysis tool puts formulas in the worksheet. Cell BS is high- lighted and the formula = 0.6 X A4 + 0.4 X B4 is shown on the formula toolbar.

162 CHAPTER 4 Moving A verages and Smoothing Meth ods

Exponential Smoothing X

Input

Input Range:

Q_amping factor:

R: Labels

_il OK ] j$A$Jj;~$25 J Jo~ _J Cancel

t!elp I

New Worksheet Ply:

2.tandard Errors

FIGURE 4-25 Excel Exponential Smoothing Dialog Box

• .J.QJE ~ fie t_dl: ~- Insert f2,mat Iools Qal:• l'!)idow !:jelp 1ype a q.,est1on for~

~Q B eat ~ l~ :t:I. i ttl - ~ J ") - t" - l ~ 1:- H l !Vl !l1 Jl ' -i,5oa0ltt?l'l'mdow

• - fJ X

\ Arial , ~--~ 11". i1" -= - &·,A,·J '~---~- -- 85

G H J

Exponential Smoothing

--+- Actual

1 4 7 10 13 16 19 22

Data Point

FIGURE 4-26 Excel Exponential Smoothing: Example 4.5 Results

5. Notice that, althou~h Excel s tarts exponential smoothing differe ntly_froID the way the smooth_mg was started in Table 4-7 (see the first few values ID t~e column corres pondmg to a = .6) , after six or seven iterations the numbers UI

CHAPTER 4 Moving Averages and Smoothing Methods 1 63

colum~ B in Figure 4-26 and the numbers in the column corresponding to a = .6 m Table 4-7 are exactly the same. Fluctuations due to different choices of starting values die out fairly rapidly .

References

Aaker, D. A., and R. Jacobson. "The Sophistication of 'Naive' Modeling." International Journal of Forecasting 3 (314) (1987): 449-452.

Bowerman, B. L. , R. T. O'Connell, and A. B. Koehler. Forecasting, Time Series and R egression, 4th ed. Belmont, CA: Thomson Brooks/Cole, 2005.

Dalrymple, D. J. , and B . E. King. " Selecting Parameters for Short-Term Forecasting Techniques." Decision Sciences 12 (1981): 661-669.

Gardner, E . S., Jr. " Exponential Smoothing:The State of the Art." Journal of Forecasting 4 (1985): 1-28.

Gardner, E. S. , Jr. , and D. G. Dannenbring . "Forecasting with Exponential Smoothing: Some Guidelines for Model Selection." Decision Sciences 11 (1980) : 370--383.

Holt, C. C. "Forecasting Seasonals and Trends by Exponentially Weighted Moving Averages." International Journal of Forecasting 20 (2004): 5-10.

Holt, C. C., F. Modigliani, J. F. Muth, and H . A. Simon. Planning Production Inventories and

Work Force. Englewood Cliffs, NJ.: Prentice-Hall, 1960.

Koehler, A. B., R. D. Snyder, and D. K. Ord. " Forecasting Models and Prediction Intervals for the Multiplicative Holt-Winters Method." International Journal of Forecasting 17 (2001): 269-286.

Ledolter, J. , and B. Abraham. "Some Comments on the Initialization of Exponential Smoothing." Journal of Forecasting 3 (1) (1984): 79--84.

Makridakis, S., S. C. Wheelwright, and R. Hyndman. Forecasting Methods and Applications. New York: Wiley, 1998.

McKenzie, E . "An Analysis of General Exponential Smoothing." Operations Research 24 (1976): 131-140.

Newbold, P. , and T. Bos. Introductory Business and Economic Forecasting, 2nd ed. Cincinnati, Ohio: South-Western, 1994.

Winters, P. R. "Forecasting Sales by Exponentially Weighted Moving Averages." Management Science 6 (1960): 324-342.