- Email: [email protected]
Fitting exponential smoothing models with computer graphics
Computers ind. Engng Vol.15, Nos 1-4, pp.369-377, 1988 Printed in Great B r i t a i n .
FITTING
All r i g h t s reserved
EXPONENTIAL
Rafael
SMOOTHING
G. Moras
0360-8352/88 $3.00+0.00 Copyright c 1988 Pergamon Press plc
MODELS WITH COMPUTER
GRAPHICS
and Eric L. Blair
Department of Industrial Engineering Texas Tech University Lubbock, TX 79409
INTRODUCTION
The program performs several types of exponential smoothing forecasts. Exponential s m o o t h i n g uses all the p r e v i o u s data points a d j u s t i n g the weights given to data so that older points are given increasingly less weight. This data m a n i p u l a t i o n process is c a r r i e d out in a way which requires only the most recent results to be saved. E x p o n e n t i a l smoothing forecasts have an intuitive appeal in that t r e n d and seasonal factors can be i n c o r p o r a t e d into the model so that a more realistic forecast is produced.
The relevance of applying forecasting techniques in m a n a g i n g a p r o d u c t i o n system is unquestionable. In fact, the m a j o r i t y of the d e c i s i o n s made in a c o m p a n y are based on a forecast, a n d the q u a l i t y of t h o s e d e c i s i o n s u s u a l l y reflects the quality of the forecast. W h i l e the f o r m u l a s employed in forecasting can be easily understood by most users, the development of an a c c e p t a b l e forecast is almost an art. Among the various available methods (and c o m b i n a t i o n s of such methods), it is d e s i r a b l e to c h o o s e the one that produces the closest m a t c h between the forecast and the actual data. P r e d i c t i o n s of future p e r f o r m a n c e of a system can be made with a higher level of c o n f i d e n c e if the past has b e e n m a t c h e d with reasonable accuracy.
THE BASIC LEVEL MODEL Let Xi be the v a l u e of past data, where "i" indicates the p e r i o d in which an o b s e r v a t i o n occurred. Yn+l, the forecast for p e r i o d n+l, is equal to the weighted sum of all the previous observations:
The q u a l i t y of a forecast can be measured by quantitative and qualitative means. Among the former are the c o r r e l a t i o n coefficient a n d the mean absolute deviation, which provide a numerical basis to e v a l u a t e the forecast. A q u a l i t a t i v e assessment can be a c c o m p l i s h e d by v i s u a l l y e x a m i n i n g the forecast and the actual data. A combination of the a b o v e m e t h o d s is usually r e q u i r e d to p r o d u c e a valid evaluation of the forecast.
Y.+I = S = ~ X where
~ is
the
+ (l-~)S. I
(i)
exponential
smoothing constant (0<~
A p r o g r a m called EXPOFIT has been developed and is r e p o r t e d in t h i s study. E X P O F I T p r o v i d e s the user with the opportunity to try several f o r e c a s t i n g schemes on a data set. The p r o g r a m takes advantage of the graphic c a p a b i l i t i e s of the m i c r o - c o m p u t e r to enhance the h i s t o r y - m a t c h i n g process by p r o v i d i n g a plot in w h i c h the actual h i s t o r y a n d r e s u l t s p r o d u c e d by the p r o p o s e d forecast are compared. It also f u r n i s h e s the m e a n a b s o l u t e d e v i a t i o n and the correlation coefficient for z each model. E X P O F I T is w r i t t e n in MS i -BASIC for i m p l e m e n t a t i o n on IBM, TI, and compatible microcomputers. While the c o m p u t a t i o n routines are identical in b o t h v e r s i o n s , m i n o r c h a n g e s are necessary to implement the graphic routines on each machine.
105; and Weights of X14 and S13 forecast is
the value of ~ is 0.2. 0.2 and 0.8 are assigned to respectively. The resulting obtained by the formula:
Y,5 = S14 = 0"2X14 + 0"8S13 •117 AS soon as X15 occurs, the n e w forecast, Y16 can be calculated. Only X15 a n d $14 are n e e d e d to p r o d u c e the
369
370
Proceedings of the 10th Annual Conference on Computers & Industrial Engineering
n e w forecast. S i n c e in this m o d e l it is assumed that no trend or seasonal forces exist, the forecast for k observations i n t o the f u t u r e is e q u a l to the last forecast.
The f o r e c a s t for the next p e r i o d is m o d i f i e d to i n c l u d e t h e s e a s o n a l c o m p o n e n t in the f o l l o w i n g manner: /Cn-p+l
Vn+l= ~nt--'~n ) T~END
(6)
ENHANCEMENT
The possibility that a trend a f f e c t s t h e d a t a is c o n s i d e r e d as an enhancement to t h e b a s i c model. A smoothed estimate of the trend is i n c l u d e d in the f o r e c a s t so t h a t the smoothed average, Sn, w i l l f o l l o w the trend process. The forecast for t h e next p e r i o d is d e f i n e d as:
W h e n b o t h the t r e n d a n d s e a s o n a l c o m p o n e n t s are i n c l u d e d in the model, the f o r e c a s t is d e f i n e d as:
Y-+x = (Sn + 1Tn (X )
Cn-l+l C
(7)
STATISTICS (2)
Yn+l = Sn + I Tn
w h e r e Tn is the t r e n d c o m p o n e n t , w h i c h can a l s o be s m o o t h e d w i t h the a d d i t i o n of the trend smoothing constant (0<~
(3)
"In = 8(Sn-Snq) + (1-B)Tnq
w h e r e Tn is a s m o o t h e d a v e r a g e of the current and previous t r e n d s . As w i t h the b a s i c f o r e c a s t i n g model, it is o n l y n e c e s s a r y to save the v a l u e of the last e s t i m a t e of the t r e n d c o m p o n e n t .
the
The forecast for k p e r i o d s f u t u r e is f o u n d as follows:
Yn+k = Sn + (k - I + I ) Tn
SEASONAL
into
(4 )
ENHANCEMENT
A desirable extension to the previous models is a m o d e l in w h i c h cyclical or seasonal patterns are i n c o r p o r a t e d in the f o r e c a s t . S e a s o n a l t y p e f o r e c a s t s are u s e f u l w h e n it is believed t h a t t h e f o r c e s f o r m i n g the h i s t o r i c a l d a t a c h a n g e f r o m p e r i o d to p e r i o d and are r e p l i c a t e d in a s i m i l a r fashion in t h e n e x t c y c l e . The base forecast is m o d i f i e d to take into account seasonal deviations f r o m the average. In this paper, winter's m e t h o d (as d e s c r i b e d in [I]) is u s e d in the derivation of t h e forecast. A s e a s o n a l c o m p o n e n t Cn is d e f i n e d as:
X
Cn=~(~nn) +(1-q)Cn. p where
p
is
the
number
of
(5) periods
in a c y c l e , a n d y is the seasonal s m o o t h i n g c o n s t a n t . The s e a s o n a l i n d e x C~ p r o v i d e s an e s t i m a t e of h o w m u c h the d e m a n d d u r i n g a s e a s o n will be a b o v e or b e l o w the a v e r a g e demand.
EXPOFIT provides statistical i n d i c e s w h i c h can be u s e d to a s s e s s the q u a l i t y of a forecast. The c o r r e l a t i o n coefficient, r, p r o v i d e s a m e a s u r e of the s t r e n g t h of the l i n e a r r e l a t i o n s h i p b e t w e e n t w o v a r i a b l e s . In EXPOFIT, the correlation coefficient is u s e d to m e a s u r e the l i n e a r r e l a t i o n s h i p b e t w e e n historical data and the forecast of t h a t d a t a set. A v a l u e of r c l o s e to 1.0 m e a n s than there is a p o s i t i v e correlation between actual data and f o r e c a s t . A c o r r e l a t i o n c o e f f i c i e n t of 0.0 i n d i c a t e s no fit. A n e g a t i v e v a l u e of r r e s u l t s w h e n the d e m a n d a n d the forecast take opposite directions, i.e., when the demand is high, the forecast is low. Correlation c o e f f i c i e n t v a l u e s c l o s e to 1.0 s u g g e s t a g o o d fit. O n e of the p r o b l e m s of u s i n g this s t a t i s t i c a l tool as the only m e a s u r e of p e r f o r m a n c e is that in some c a s e s it w i l l not d e t e c t a b i a s in t h e forecast. For instance, it c o u l d be p o s s i b l e to o b t a i n a h i g h c o r r e l a t i o n c o e f f i c i e n t w h e n m o s t of the f o r e c a s t p o i n t s are s m a l l e r t h a n a c t u a l h i s t o r y but follow the same trend and/or cycles. The correlation c o m p u t e d as follows:
r
=
i=l
coefficient
is
(xi " x ) (Yi - )7 ) (8)
N~/~(xi" ~ ) 2 ~ i = 1 i=1 (Yi"Y)2 w h e r e xi r e p r e s e n t s the a c t u a l history, Yi is the c o r r e s p o n d i n g forecast, a n d n is the n u m b e r of p o i n t s for w h i c h a f o r e c a s t is p r o d u c e d .
Horas and Blair : Exponential ssoothlng models
The mean absolute deviation (MAD) is an alternative measure of v a r i a b i l i t y that has greater acceptance mainly because of its c o m p u t a t i o n a l simplicity. The M A D is c o m p u t e d as follows:
for h i s t o r y (t):
[5].
The starting values of the base, trend, and seasonal values usually play an important role in fitting data with an e x p o n e n t i a l s m o o t h e d forecast. The q u a l i t y of the i n i t i a l e s t i m a t e s is normally r e f l e c t e d in the quality of the forecast, e s p e c i a l l y when the data set c o n s i s t s of a l i m i t e d n u m b e r of periods. In general, when the data set is s u f f i c i e n t l y large, the effect of poor initial estimates is e v e n t u a l l y s m o o t h e d out. Thus the p e r f o r m a n c e of the i n i t i a l i z a t i o n routine is u s u a l l y more critical w h e n the d a t a set is small. E X P O F I T p r o v i d e s the user with the o p p o r t u n i t y to vary the length of the initialization interval and observe h o w this a f f e c t s the q u a l i t y of the forecast. The i n i t i a l i z a t i o n routines for the basic model, the trend -enhanced model, and the seasonal model are d e s c r i b e d below. Let n be the n u m b e r of p e r i o d s used in the i n i t i a l i z a t i o n p e r i o d (as i n d i c a t e d by the user). In the basic model, the estimate for the base value is o b t a i n e d by computing the average of the first n points of history:
~-
(i0)
i=l
This procedure
is suggested
in
[i].
In the t r e n d e n h a n c e d model, the estimates for Sn and Tn are obtained by applying linear regression to the first n points of history. The application of linear regression p r o v i d e s an equation
of time
(11)
of Tn is set equal to b:
(9)
In this way, the average magnitude of the e r r o r is c o m p u t e d i r r e s p e c t i v e of w h e t h e r it is positive or negative. Generally, low values of MAD indicate acceptable forecasts, although it is a l w a y s important to compare the relative value of MAD with the a v e r a g e size of the data. A MAD of 5.0 w o u l d be c o n s i d e r e d excellent for data w i t h an a v e r a g e v a l u e of i00.0, but it w o u l d p r o b a b l y be c o n s i d e r e d an i n d i c a t o r of a poor fit for data with an a v e r a g e v a l u e of 0.8. An e x t e n d e d description of e x p o n e n t i a l smoothing can be found in [i], [2], [3], [4], and
a function
where a is the value of X at t ffi 0, and b is the slope of the line. The value
i=I
Sn=
as
Xfa+~
~,Ix~-Yil MAD=
(X)
371
Tn=b.
(12)
The following equation is d e r i v e d from Eq. II and is used in E X P O F I T to find the initial estimate of Sn:
Sn=a+bn. This C1].
procedure
is
also
a b
l-a
suggested
(13) in
In the seasonal model the length of the initialization period is restricted to be a m u l t i p l e of t h e number of seasons per cycle. The estimation for the b a s e value, the t r e n d value, and the seasonal factors starts by f i n d i n g the seasonal factor for each season. The task is p e r f o r m e d for each cycle:
pX~ =__t X.
CS~ =
P
X Xi
(14)
X
P
w h e r e CSkl is the seasonal factor s e a s o n i in c y c l e k, a n d Xp is average value in that cycle.
for the
The estimate for the average seasonal factor, Ci, is c o m p u t e d by t a k i n g the a v e r a g e of the CSki values for e a c h p e r i o d i, over the e n t i r e initialization period: K
X CS~ Ci = k = I
(15)
K where K is the number of cycles in the initialization period. EXPOFIT uses the average seasonal factors over the i n i t i a l i z a t i o n p e r i o d as the s t a r t i n g estimates for Ci. E a c h d e m a n d value is d i v i d e d by its c o r r e s p o n d i n g seasonal factor CSkl to obtain a "deseasonalized" X± value. The average and/or t r e n d estimates are then calculated based on the transformed data. The i n i t i a l i z a t i o n procedure for a s e a s o n a l model is illustrated with the following example.
372
Proceedings of the 10th Annual Conference on Computers & Industrial Eoglneerfng
Initialization
for a Seasonal
the m o d e l "form"). The second step is to determine the values which should be used for the "smoothing constants" (or w e i g h t i n g factors ~.~,andy) .
Model.
Assume that the first 8 points of the data set in T a b l e 1 are u s e d to obtain the initial estimates for a s e a s o n a l - e n h a n c e d model with trend. It is a s s u m e d that each cycle has 4 seasons. The a v e r a g e v a l u e s for the first and second cycles are 14 a n d 16.5, respectively, which are used to o b t a i n the s e a s o n a l factors CSki for each cycle. The average seasonal factor for the first season is c o m p u t e d as follows:
C1=
The s e l e c t i o n of a m o d e l form should reflect both obvious visual patterns in past data and the analyst's knowledge of the real w o r l d p r o c e s s w h i c h is g e n e r a t i n g the time series b e i n g forecast. One of the simplest and most r e v e a l i n g analysis procedures is to plot the h i s t o r i c a l data versus time. Strong trend and seasonal effects are easily identified from the plot as are the a b s e n c e of such conditions. Knowledge of the type and source of the data is a l s o an i m p o r t a n t factor in determining the model form. The existence of cycles of certain periods may be a n t i c i p a t e d to occur because of the u n d e r l y i n g process. For example, m o n t h l y sales of g a s o l i n e m a y show a strong seasonal trend with above average consumption in the summer months. W e e k l y gasoline sales may hide the s e a s o n a l e f f e c t s b e c a u s e of the i n c r e a s e d n u m b e r of periods per cycle (52 i n s t e a d of 12) and the impact of white noise (we s h o u l d e x p e c t more relative variation w i t h w e e k l y data than with monthly data).
CSH + CS12 0.857 + 0.909 = 0.883 2 = 2
The remaining average seasonal are computed in a similar way:
factors
CI = 0.883
C2 = 1.021 C 3 = 1.116
C4 = 0.980 The data set is " d e s e a s o n a l i z e d " by dividing each point by its corresponding Ci, thus o b t a i n i n g the v a l u e s on c o l u m n (6), w h i c h in turn w i l l be u s e d to c o m p u t e this initial estimates for the base and trend values. For this example, the initial b a s e a n d t r e n d values are found to be 15.007 a n d 0.491 respectively.
CHOOSING
A SMOOTHING
S e l e c t i o n of s m o o t h i n g constants is o f t e n t h o u g h t of as a p r o c e s s of "fitting the model to data". In fact, this is d e f i n i t e l y not the case. The choice of smoothing constant values is a management decision regarding the tradeoff between two inconsistent f o r e c a s t i n g objectives: (i) s m o o t h i n g out n o i s e in p r o c e s s g e n e r a t e d data, and (2) p r o v i d i n g a quick r e c o g n i t i o n of a real sustained change in the form of the g e n e r a t i n g process. As such, this d e c i s i o n has little to do with fitting a curve to data.
CONSTANT
C h o o s i n g an e x p o n e n t i a l smoothing m o d e l i n v o l v e s two d i s t i n c t d e c i s i o n processes. The first step is to d e t e r m i n e the a p p r o p r i a t e n e s s of t r e n d a n d / o r seasonal factors (i.e., choose
(I) Period
(2) X
(3)
(4)
(5)
(6)
k
i
CS~
X/Ci
First Cycle
1 2 3 4
12 15 16 13
1 1 1 1
1 2 3 4
0.857 1.071 1.143 0.929
13.59 14.69 14.34 13.27
Second Cycle
5 6 7 8
15 16 18 17
2 2 2 2
1 2 3 4
0.909 0.970 1.091 1.030
16.99 15.67 16.13 17.35
Third Cycle
9 10 11 12
15 17 17 16
3 3 3 3
1 2 3 4
Table
I.
Calculation
of seasonal
factors.
Moras and B l a i r
:
Exponential smoothing models
The role of the smoothing constants is to determine the relative importance, or weight, given to "new information" (most recent data) versus "old information" (previously smoothed values for level, t r e n d and seasonal elements of t h e forecast). A high value of a smoothing constant will give more importance to new information and less i m p o r t a n c e to o l d i n f o r m a t i o n . Consequently, when new information indicates that the process has changed, the forecast is capable of recognizing the shift. This is shown in Fig. i, where a level smoothing model is being u s e d to forecast for a level p r o c e s s which has e x p e r i e n c e d a sudden rise in
period
21.
5=0.3
is
The much
373
resulting more
forecast
responsive
that for 5=0.1. The d o w n s ! d e to this responsiveness property is that a f o r e c a s t i n g model with high values for its smoothing constants will not recognize real changes from noise and cannot do a very good job of smoothing it out of the forecast. This is illustrated in Fig. 2, where a noisy, but persistent process has been smoothed by two level models with 5=0.1 and ~=0.3. The model with 5 = 0 . 1 does a much better job of smoothing out the n o i s e about the p r o c e s s mean of 120.
50 40 30 20
~03 I
10
!
0
Figure
alpha = 0.1
!
10
!
20 i.
Sudden
40
30 shift
in d a t a .
160
T .,stor
/
140
Alpha = 0.3 Alpha = 0.1
120 100 80
i
0
Figure C&Z|~S:I/4-¥
I
10
20 2.
for than
Smoothing
I
I
30 noise
40 data.
50
374
Proceedings of the 10th Annual Conference on Computers & Industrial Engineering
The determination of s m o o t h i n g constant values, analogous to m o d e l selection, is s i g n i f i c a n t l y a i d e d by the use of g r a p h i c s t e c h n i q u e s in w h i c h the r e s u l t s of s e v e r a l a l t e r n a t i v e s can be directly compared. When such procedures must be carried out manually, the t a s k b e c o m e s s l o w a n d onerous. Clearly, this presents a good opportunity for the use of a microcomputer graphics technique.
USING
THE P R O G R A M
E X P O F I T is a m e n u - d r i v e n , user friendly program. A f t e r l o a d i n g the program, t h e u s e r has the o p t i o n of r e a d i n g a d a t a set f r o m the d i s k or entering d a t a f r o m the k e y b o a r d . The i n i t i a l m e n u p r e s e n t e d to the u s e r is d e p i c t e d in Fig. 3.
from the analysis "RETURN" ("ENTER") the f i t t i n g r o u t i n e 5.
FITflNG ROUTINE Znter linear smoothing constant (alpha) Enter the trend smoothing constant (beta) Enter the seasonal' constant (gamma) Enter the number Of periods per cycle5 View FIT calculations Yes or No Clear plot arrays Yes or No
I. 2. 3. 4. 5. 6. 7.
Enter data from keyboard Store data on disk Load stored data V i e w / a l t e r data Fit/plot data Forecast Quit
WXICH OPTION DO YOU DESIRE?
Fig.
3.
Initial
.2 .3 .I Y N
Enter the number of periods to be used in the Inltializatlon routlne (A multiple of the number of perlods per cycle 5, I0, 15 .... )
Fig.
YOU HAVE THE FOLLOWING OPTIONS:
by p r e s s i n g the key. The m e n u for is d e p i c t e d in Fig.
5. F i t t i n g
Routine
E X P O F I T p r o v i d e s a l i s t i n g of the values of the linear, trend, and seasonal factors (denoted by SBAR, TBAR, and CBAR in the program) as t h e y are g e n e r a t e d . This listing includes the forecast produced at t h e e n d of each period. An e x a m p l e of t h i s is • p r o v i d e d in Fig. 6.
Perlod 31 32 33 34
X 35.00 33.00 37.00 40.00
SBRR 25.64 27.11 29.09 31.27
TBAR 1.58 1.56 1.64 1.75
CBAR FORECAST 1.29 26.38 1.05 38.70 1.19 40.51 1.29 32.78
Menu
The d a t a set can be l i s t e d a n d m o d i f i e d at any time. (See Fig. 4). It is a l s o p o s s i b l e to l o a d a n e w d a t a set, to save the c u r r e n t d a t a set, a n d to o b t a i n a p l o t of the d a t a b e f o r e a fit is a t t e m p t e d .
< PRESS ANY KEY TO CONTINUE >
MODIFYING DATA Fig.
OBS. No I 2 3 4 5
Xfll I0.000 23.335 25.453 29.443 21.110
CHANGE DATA? Y)es or N)o OBS. No._4___
Figure
NEW VALUE ? 19.443
4. D a t a M o d i f i c a t i o n
The user can choose to fit an e x p o n e n t i a l s m o o t h i n g m o d e l to the data by s p e c i f y i n g a c o m b i n a t i o n of v a l u e s for t h e s m o o t h i n g constants. While a s m o o t h i n g c o n s t a n t for the b a s e v a l u e is always required, the trend and seasonal components can be e x c l u d e d
6. Fit
Calculations
The use of the E X P O F I T p r o g r a m is i l l u s t r a t e d w i t h the f o l l o w i n g example. It is d e s i r e d to f i n d the c o m b i n a t i o n of smoothing constants which would y i e l d an a p p r o p r i a t e fit b e t w e e n the f o r e c a s t a n d the a c t u a l v a l u e s for the d a t a p l o t t e d in Fig. 7. This d a t a set appears to f o l l o w a r i s i n g t r e n d in conjunction with a 12-period seasonal p a t t e r n . In t h i s case a l p h a is g i v e n a s t a r t i n g v a l u e of 0.I. The t r e n d a n d s e a s o n a l s m o o t h i n g c o n s t a n t s are first d e f a u l t e d to zero. E X P O F I T a l l o w s the u s e r to e x a m i n e the c o m p u t e d v a l u e s of the base, trend, and seasonal values (denoted b y SBAR, TBAR, and CBAR in the program), and the forecast generated for e a c h p e r i o d . The c a l c u l a t i o n s are f o l l o w e d b y the p l o t t i n g routine.
Moras and B l a i r
: E x p o n e n t i a l smoothing models
375
90 80 70 60 50 40 30
e
0
|
10 Figure
7.
i
20 Data
with
e
30 trend
Since finding an acceptable forecast is a process that may require several iterations, the user has the o p t i o n of t r y i n g d i f f e r e n t values of the smoothing constants until a satisfactory match is achieved. One of the most useful features of EXPOFIT is that the most recent forecast can be compared with previous forecasts by using a m u l t i c o l o r e d plot. This feature provides a means of e s t i m a t i n g the s e n s i t i v i t y of the r e s u l t i n g forecast with respect to changes in the smoothing constants. The resulting plot can be d i s c a r d e d at any time. The superimposed plots for .the o r i g i n a l history and the basic forecast obtained with ~ = 0.i are shown in Fig. 8.
and
40 seasonal
50
patterns.
To show h o w the b a s i c model can be e n h a n c e d , s u p p o s e that t r e n d and seasonal smoothing components are included in the analysis, in this case with values of ~=0.20, 5=0.20, ~=0.25, and 12 periods per cycle. The resulting forecast is shown in Fig. 9. T h e enhanced model appears to y i e l d a closer fit than the basic model. The user may wish to try different schemes until a satisfactory match is achieved. Once an acceptable match has been ~obtained, the user can request a f o r e c a s t for k periods into the future. Since a forecast usually loses accuracy when it is u s e d to p r e d i c t l o n g - t e r m behavior, this feature should be used with caution. Continuous review of the smoothing parameters is a d v i s a b l e so that changing conditions can be accurately reflected in the model.
90 8O 70 60 5O 40 30 20
-". 10
0 Figure
8.
.
. 20
Base
model
trend
and
History
.
" 30
smoothing seasonal
40 of
data
patterns.
with
50
376
Proceeding8 of the lOth Annual Conference on Computer8 & Industrial
9O 8O 70 60 5O 40 30 2O 0
Trend & Season I
I" i
i
10 Figure
i
20 9.
Engineering
i
30
40
Enhanced smoothing t r e n d a n d seasonal
of d a t a
50
with
factors.
CONCLUDING REMARKS The p r o g r a m p r e s e n t e d in this paper provides a valuable tool in selecting the most adequate forecasting method. The usually tedious trial-anderror history matching approach is highly simplified by the speed and ease of use the m i c r o c o m p u t e r and by the v i s u a l aids p r o v i d e d by the program. E X P O F I T is e x t r e m e l y flexible in that different matching schemes can be i m p l e m e n t e d by c h a n g i n g the values of the smoothing constants. The effect of these changes can be analyzed i n d i v i d u a l l y or in c o n j u n c t i o n with previous results. By e x a m i n i n g the superimposed plots, the user can develop an intuitive sense of what the most appropriate combination of parameters should be. Thus the inevitable t r i a l - a n d - e r r o r process may become f a s t e r and less t e d i o u s . In addition to the graphical aids, EXPOFIT provides the u s e r with s t a t i s t i c a l standards commonly used to evaluate the q u a l i t y of the forecast, such as the c o r r e l a t i o n c o e f f i c i e n t and the mean absolute deviation. E X P O F I T also allows the user to try d i f f e r e n t i n i t i a l i z a t i o n schemes and analyze the effect of these a l t e r n a t i v e s on the resulting forecast. Initialization is e x t r e m e l y important when the n u m b e r of data p o i n t s is limited. The p r o g r a m takes a d v a n t a g e s of the user-friendliness of microcomputers. In addition to being a valid m e a n s to e v a l u a t e the q u a l i t y of a forecast, EXPOFIT furnishes an excellent methodology for the evaluation of the responses of a forecasting model to changes in historical data.
[i]
E. A. Silver and R. Peterson,
Decision Systems for Inventory Management and Production Planning, Second Edition, Wiley, (1985). [2]
T. E. Vollmann, W. L. Berry, D. C. Whybark, Manufacturing
and
Planning and Control Systems, Irwin, [3]
R. G. Brown, Smoothing, Forecasting and Prediction of Discrete Time Series, PrenticeHall,
[4]
(1984).
{1963).
A. C. Hax
and D. Candea,
Production and Inventory Management, Prentice Hall, [5]
(1984)
D. D. Bedworth and J. E. Bailey,
Integrated Production Control Systems. Management, Analysis, Design, Second Edition, Wiley, (1987). [6]
P. J. Harrison, Exponential Smoothing and Short-Term Sales
Forecasting, Management Science, 13, [7]
821-842
(1967).
T. D. Russell and E. E. Adam, An Empirical Evaluation of Alternative Forecasting Combinations, Management Science, 15, 1267-1276 (1987).
Moras and B l a i r
BIOGRAPMICAL
:
Exponentlal smoothing models
SMETCMES
Dr. R a f a e l G. Moras, is an a s s i s t a n t professor at the Department of Industrial Engineering, Texas Tech University. He received a BS in Industrial and Systems Engineering from Monterrey Institute of T e c h n o l o g y , Mexico, a MSE f r o m the U n i v e r s i t y of Texas at Austin, and a PhD also from UT Austin. His current research interests include applications of operations research to problems in p r o d u c t i o n and inventory systems. Dr. Moras is a member of TIMS, ORSA, and IIE. Dr. E r i c L. Blair, is an a s s o c i a t e professor at the Department of Industrial Engineering, Texas Tech University. He received a BS in Management Engineering a n d a PhD in Operations Research and Statistics from Rensselaer Polytechnic Institute. His current research interests include computer simulation, routing and logistics, and quality insurance. He is a member of TIMS, ORSA, and IIE.
377
FITTING
All r i g h t s reserved
EXPONENTIAL
Rafael
SMOOTHING
G. Moras
0360-8352/88 $3.00+0.00 Copyright c 1988 Pergamon Press plc
MODELS WITH COMPUTER
GRAPHICS
and Eric L. Blair
Department of Industrial Engineering Texas Tech University Lubbock, TX 79409
INTRODUCTION
The program performs several types of exponential smoothing forecasts. Exponential s m o o t h i n g uses all the p r e v i o u s data points a d j u s t i n g the weights given to data so that older points are given increasingly less weight. This data m a n i p u l a t i o n process is c a r r i e d out in a way which requires only the most recent results to be saved. E x p o n e n t i a l smoothing forecasts have an intuitive appeal in that t r e n d and seasonal factors can be i n c o r p o r a t e d into the model so that a more realistic forecast is produced.
The relevance of applying forecasting techniques in m a n a g i n g a p r o d u c t i o n system is unquestionable. In fact, the m a j o r i t y of the d e c i s i o n s made in a c o m p a n y are based on a forecast, a n d the q u a l i t y of t h o s e d e c i s i o n s u s u a l l y reflects the quality of the forecast. W h i l e the f o r m u l a s employed in forecasting can be easily understood by most users, the development of an a c c e p t a b l e forecast is almost an art. Among the various available methods (and c o m b i n a t i o n s of such methods), it is d e s i r a b l e to c h o o s e the one that produces the closest m a t c h between the forecast and the actual data. P r e d i c t i o n s of future p e r f o r m a n c e of a system can be made with a higher level of c o n f i d e n c e if the past has b e e n m a t c h e d with reasonable accuracy.
THE BASIC LEVEL MODEL Let Xi be the v a l u e of past data, where "i" indicates the p e r i o d in which an o b s e r v a t i o n occurred. Yn+l, the forecast for p e r i o d n+l, is equal to the weighted sum of all the previous observations:
The q u a l i t y of a forecast can be measured by quantitative and qualitative means. Among the former are the c o r r e l a t i o n coefficient a n d the mean absolute deviation, which provide a numerical basis to e v a l u a t e the forecast. A q u a l i t a t i v e assessment can be a c c o m p l i s h e d by v i s u a l l y e x a m i n i n g the forecast and the actual data. A combination of the a b o v e m e t h o d s is usually r e q u i r e d to p r o d u c e a valid evaluation of the forecast.
Y.+I = S = ~ X where
~ is
the
+ (l-~)S. I
(i)
exponential
smoothing constant (0<~
A p r o g r a m called EXPOFIT has been developed and is r e p o r t e d in t h i s study. E X P O F I T p r o v i d e s the user with the opportunity to try several f o r e c a s t i n g schemes on a data set. The p r o g r a m takes advantage of the graphic c a p a b i l i t i e s of the m i c r o - c o m p u t e r to enhance the h i s t o r y - m a t c h i n g process by p r o v i d i n g a plot in w h i c h the actual h i s t o r y a n d r e s u l t s p r o d u c e d by the p r o p o s e d forecast are compared. It also f u r n i s h e s the m e a n a b s o l u t e d e v i a t i o n and the correlation coefficient for z each model. E X P O F I T is w r i t t e n in MS i -BASIC for i m p l e m e n t a t i o n on IBM, TI, and compatible microcomputers. While the c o m p u t a t i o n routines are identical in b o t h v e r s i o n s , m i n o r c h a n g e s are necessary to implement the graphic routines on each machine.
105; and Weights of X14 and S13 forecast is
the value of ~ is 0.2. 0.2 and 0.8 are assigned to respectively. The resulting obtained by the formula:
Y,5 = S14 = 0"2X14 + 0"8S13 •117 AS soon as X15 occurs, the n e w forecast, Y16 can be calculated. Only X15 a n d $14 are n e e d e d to p r o d u c e the
369
370
Proceedings of the 10th Annual Conference on Computers & Industrial Engineering
n e w forecast. S i n c e in this m o d e l it is assumed that no trend or seasonal forces exist, the forecast for k observations i n t o the f u t u r e is e q u a l to the last forecast.
The f o r e c a s t for the next p e r i o d is m o d i f i e d to i n c l u d e t h e s e a s o n a l c o m p o n e n t in the f o l l o w i n g manner: /Cn-p+l
Vn+l= ~nt--'~n ) T~END
(6)
ENHANCEMENT
The possibility that a trend a f f e c t s t h e d a t a is c o n s i d e r e d as an enhancement to t h e b a s i c model. A smoothed estimate of the trend is i n c l u d e d in the f o r e c a s t so t h a t the smoothed average, Sn, w i l l f o l l o w the trend process. The forecast for t h e next p e r i o d is d e f i n e d as:
W h e n b o t h the t r e n d a n d s e a s o n a l c o m p o n e n t s are i n c l u d e d in the model, the f o r e c a s t is d e f i n e d as:
Y-+x = (Sn + 1Tn (X )
Cn-l+l C
(7)
STATISTICS (2)
Yn+l = Sn + I Tn
w h e r e Tn is the t r e n d c o m p o n e n t , w h i c h can a l s o be s m o o t h e d w i t h the a d d i t i o n of the trend smoothing constant (0<~
(3)
"In = 8(Sn-Snq) + (1-B)Tnq
w h e r e Tn is a s m o o t h e d a v e r a g e of the current and previous t r e n d s . As w i t h the b a s i c f o r e c a s t i n g model, it is o n l y n e c e s s a r y to save the v a l u e of the last e s t i m a t e of the t r e n d c o m p o n e n t .
the
The forecast for k p e r i o d s f u t u r e is f o u n d as follows:
Yn+k = Sn + (k - I + I ) Tn
SEASONAL
into
(4 )
ENHANCEMENT
A desirable extension to the previous models is a m o d e l in w h i c h cyclical or seasonal patterns are i n c o r p o r a t e d in the f o r e c a s t . S e a s o n a l t y p e f o r e c a s t s are u s e f u l w h e n it is believed t h a t t h e f o r c e s f o r m i n g the h i s t o r i c a l d a t a c h a n g e f r o m p e r i o d to p e r i o d and are r e p l i c a t e d in a s i m i l a r fashion in t h e n e x t c y c l e . The base forecast is m o d i f i e d to take into account seasonal deviations f r o m the average. In this paper, winter's m e t h o d (as d e s c r i b e d in [I]) is u s e d in the derivation of t h e forecast. A s e a s o n a l c o m p o n e n t Cn is d e f i n e d as:
X
Cn=~(~nn) +(1-q)Cn. p where
p
is
the
number
of
(5) periods
in a c y c l e , a n d y is the seasonal s m o o t h i n g c o n s t a n t . The s e a s o n a l i n d e x C~ p r o v i d e s an e s t i m a t e of h o w m u c h the d e m a n d d u r i n g a s e a s o n will be a b o v e or b e l o w the a v e r a g e demand.
EXPOFIT provides statistical i n d i c e s w h i c h can be u s e d to a s s e s s the q u a l i t y of a forecast. The c o r r e l a t i o n coefficient, r, p r o v i d e s a m e a s u r e of the s t r e n g t h of the l i n e a r r e l a t i o n s h i p b e t w e e n t w o v a r i a b l e s . In EXPOFIT, the correlation coefficient is u s e d to m e a s u r e the l i n e a r r e l a t i o n s h i p b e t w e e n historical data and the forecast of t h a t d a t a set. A v a l u e of r c l o s e to 1.0 m e a n s than there is a p o s i t i v e correlation between actual data and f o r e c a s t . A c o r r e l a t i o n c o e f f i c i e n t of 0.0 i n d i c a t e s no fit. A n e g a t i v e v a l u e of r r e s u l t s w h e n the d e m a n d a n d the forecast take opposite directions, i.e., when the demand is high, the forecast is low. Correlation c o e f f i c i e n t v a l u e s c l o s e to 1.0 s u g g e s t a g o o d fit. O n e of the p r o b l e m s of u s i n g this s t a t i s t i c a l tool as the only m e a s u r e of p e r f o r m a n c e is that in some c a s e s it w i l l not d e t e c t a b i a s in t h e forecast. For instance, it c o u l d be p o s s i b l e to o b t a i n a h i g h c o r r e l a t i o n c o e f f i c i e n t w h e n m o s t of the f o r e c a s t p o i n t s are s m a l l e r t h a n a c t u a l h i s t o r y but follow the same trend and/or cycles. The correlation c o m p u t e d as follows:
r
=
i=l
coefficient
is
(xi " x ) (Yi - )7 ) (8)
N~/~(xi" ~ ) 2 ~ i = 1 i=1 (Yi"Y)2 w h e r e xi r e p r e s e n t s the a c t u a l history, Yi is the c o r r e s p o n d i n g forecast, a n d n is the n u m b e r of p o i n t s for w h i c h a f o r e c a s t is p r o d u c e d .
Horas and Blair : Exponential ssoothlng models
The mean absolute deviation (MAD) is an alternative measure of v a r i a b i l i t y that has greater acceptance mainly because of its c o m p u t a t i o n a l simplicity. The M A D is c o m p u t e d as follows:
for h i s t o r y (t):
[5].
The starting values of the base, trend, and seasonal values usually play an important role in fitting data with an e x p o n e n t i a l s m o o t h e d forecast. The q u a l i t y of the i n i t i a l e s t i m a t e s is normally r e f l e c t e d in the quality of the forecast, e s p e c i a l l y when the data set c o n s i s t s of a l i m i t e d n u m b e r of periods. In general, when the data set is s u f f i c i e n t l y large, the effect of poor initial estimates is e v e n t u a l l y s m o o t h e d out. Thus the p e r f o r m a n c e of the i n i t i a l i z a t i o n routine is u s u a l l y more critical w h e n the d a t a set is small. E X P O F I T p r o v i d e s the user with the o p p o r t u n i t y to vary the length of the initialization interval and observe h o w this a f f e c t s the q u a l i t y of the forecast. The i n i t i a l i z a t i o n routines for the basic model, the trend -enhanced model, and the seasonal model are d e s c r i b e d below. Let n be the n u m b e r of p e r i o d s used in the i n i t i a l i z a t i o n p e r i o d (as i n d i c a t e d by the user). In the basic model, the estimate for the base value is o b t a i n e d by computing the average of the first n points of history:
~-
(i0)
i=l
This procedure
is suggested
in
[i].
In the t r e n d e n h a n c e d model, the estimates for Sn and Tn are obtained by applying linear regression to the first n points of history. The application of linear regression p r o v i d e s an equation
of time
(11)
of Tn is set equal to b:
(9)
In this way, the average magnitude of the e r r o r is c o m p u t e d i r r e s p e c t i v e of w h e t h e r it is positive or negative. Generally, low values of MAD indicate acceptable forecasts, although it is a l w a y s important to compare the relative value of MAD with the a v e r a g e size of the data. A MAD of 5.0 w o u l d be c o n s i d e r e d excellent for data w i t h an a v e r a g e v a l u e of i00.0, but it w o u l d p r o b a b l y be c o n s i d e r e d an i n d i c a t o r of a poor fit for data with an a v e r a g e v a l u e of 0.8. An e x t e n d e d description of e x p o n e n t i a l smoothing can be found in [i], [2], [3], [4], and
a function
where a is the value of X at t ffi 0, and b is the slope of the line. The value
i=I
Sn=
as
Xfa+~
~,Ix~-Yil MAD=
(X)
371
Tn=b.
(12)
The following equation is d e r i v e d from Eq. II and is used in E X P O F I T to find the initial estimate of Sn:
Sn=a+bn. This C1].
procedure
is
also
a b
l-a
suggested
(13) in
In the seasonal model the length of the initialization period is restricted to be a m u l t i p l e of t h e number of seasons per cycle. The estimation for the b a s e value, the t r e n d value, and the seasonal factors starts by f i n d i n g the seasonal factor for each season. The task is p e r f o r m e d for each cycle:
pX~ =__t X.
CS~ =
P
X Xi
(14)
X
P
w h e r e CSkl is the seasonal factor s e a s o n i in c y c l e k, a n d Xp is average value in that cycle.
for the
The estimate for the average seasonal factor, Ci, is c o m p u t e d by t a k i n g the a v e r a g e of the CSki values for e a c h p e r i o d i, over the e n t i r e initialization period: K
X CS~ Ci = k = I
(15)
K where K is the number of cycles in the initialization period. EXPOFIT uses the average seasonal factors over the i n i t i a l i z a t i o n p e r i o d as the s t a r t i n g estimates for Ci. E a c h d e m a n d value is d i v i d e d by its c o r r e s p o n d i n g seasonal factor CSkl to obtain a "deseasonalized" X± value. The average and/or t r e n d estimates are then calculated based on the transformed data. The i n i t i a l i z a t i o n procedure for a s e a s o n a l model is illustrated with the following example.
372
Proceedings of the 10th Annual Conference on Computers & Industrial Eoglneerfng
Initialization
for a Seasonal
the m o d e l "form"). The second step is to determine the values which should be used for the "smoothing constants" (or w e i g h t i n g factors ~.~,andy) .
Model.
Assume that the first 8 points of the data set in T a b l e 1 are u s e d to obtain the initial estimates for a s e a s o n a l - e n h a n c e d model with trend. It is a s s u m e d that each cycle has 4 seasons. The a v e r a g e v a l u e s for the first and second cycles are 14 a n d 16.5, respectively, which are used to o b t a i n the s e a s o n a l factors CSki for each cycle. The average seasonal factor for the first season is c o m p u t e d as follows:
C1=
The s e l e c t i o n of a m o d e l form should reflect both obvious visual patterns in past data and the analyst's knowledge of the real w o r l d p r o c e s s w h i c h is g e n e r a t i n g the time series b e i n g forecast. One of the simplest and most r e v e a l i n g analysis procedures is to plot the h i s t o r i c a l data versus time. Strong trend and seasonal effects are easily identified from the plot as are the a b s e n c e of such conditions. Knowledge of the type and source of the data is a l s o an i m p o r t a n t factor in determining the model form. The existence of cycles of certain periods may be a n t i c i p a t e d to occur because of the u n d e r l y i n g process. For example, m o n t h l y sales of g a s o l i n e m a y show a strong seasonal trend with above average consumption in the summer months. W e e k l y gasoline sales may hide the s e a s o n a l e f f e c t s b e c a u s e of the i n c r e a s e d n u m b e r of periods per cycle (52 i n s t e a d of 12) and the impact of white noise (we s h o u l d e x p e c t more relative variation w i t h w e e k l y data than with monthly data).
CSH + CS12 0.857 + 0.909 = 0.883 2 = 2
The remaining average seasonal are computed in a similar way:
factors
CI = 0.883
C2 = 1.021 C 3 = 1.116
C4 = 0.980 The data set is " d e s e a s o n a l i z e d " by dividing each point by its corresponding Ci, thus o b t a i n i n g the v a l u e s on c o l u m n (6), w h i c h in turn w i l l be u s e d to c o m p u t e this initial estimates for the base and trend values. For this example, the initial b a s e a n d t r e n d values are found to be 15.007 a n d 0.491 respectively.
CHOOSING
A SMOOTHING
S e l e c t i o n of s m o o t h i n g constants is o f t e n t h o u g h t of as a p r o c e s s of "fitting the model to data". In fact, this is d e f i n i t e l y not the case. The choice of smoothing constant values is a management decision regarding the tradeoff between two inconsistent f o r e c a s t i n g objectives: (i) s m o o t h i n g out n o i s e in p r o c e s s g e n e r a t e d data, and (2) p r o v i d i n g a quick r e c o g n i t i o n of a real sustained change in the form of the g e n e r a t i n g process. As such, this d e c i s i o n has little to do with fitting a curve to data.
CONSTANT
C h o o s i n g an e x p o n e n t i a l smoothing m o d e l i n v o l v e s two d i s t i n c t d e c i s i o n processes. The first step is to d e t e r m i n e the a p p r o p r i a t e n e s s of t r e n d a n d / o r seasonal factors (i.e., choose
(I) Period
(2) X
(3)
(4)
(5)
(6)
k
i
CS~
X/Ci
First Cycle
1 2 3 4
12 15 16 13
1 1 1 1
1 2 3 4
0.857 1.071 1.143 0.929
13.59 14.69 14.34 13.27
Second Cycle
5 6 7 8
15 16 18 17
2 2 2 2
1 2 3 4
0.909 0.970 1.091 1.030
16.99 15.67 16.13 17.35
Third Cycle
9 10 11 12
15 17 17 16
3 3 3 3
1 2 3 4
Table
I.
Calculation
of seasonal
factors.
Moras and B l a i r
:
Exponential smoothing models
The role of the smoothing constants is to determine the relative importance, or weight, given to "new information" (most recent data) versus "old information" (previously smoothed values for level, t r e n d and seasonal elements of t h e forecast). A high value of a smoothing constant will give more importance to new information and less i m p o r t a n c e to o l d i n f o r m a t i o n . Consequently, when new information indicates that the process has changed, the forecast is capable of recognizing the shift. This is shown in Fig. i, where a level smoothing model is being u s e d to forecast for a level p r o c e s s which has e x p e r i e n c e d a sudden rise in
period
21.
5=0.3
is
The much
373
resulting more
forecast
responsive
that for 5=0.1. The d o w n s ! d e to this responsiveness property is that a f o r e c a s t i n g model with high values for its smoothing constants will not recognize real changes from noise and cannot do a very good job of smoothing it out of the forecast. This is illustrated in Fig. 2, where a noisy, but persistent process has been smoothed by two level models with 5=0.1 and ~=0.3. The model with 5 = 0 . 1 does a much better job of smoothing out the n o i s e about the p r o c e s s mean of 120.
50 40 30 20
~03 I
10
!
0
Figure
alpha = 0.1
!
10
!
20 i.
Sudden
40
30 shift
in d a t a .
160
T .,stor
/
140
Alpha = 0.3 Alpha = 0.1
120 100 80
i
0
Figure C&Z|~S:I/4-¥
I
10
20 2.
for than
Smoothing
I
I
30 noise
40 data.
50
374
Proceedings of the 10th Annual Conference on Computers & Industrial Engineering
The determination of s m o o t h i n g constant values, analogous to m o d e l selection, is s i g n i f i c a n t l y a i d e d by the use of g r a p h i c s t e c h n i q u e s in w h i c h the r e s u l t s of s e v e r a l a l t e r n a t i v e s can be directly compared. When such procedures must be carried out manually, the t a s k b e c o m e s s l o w a n d onerous. Clearly, this presents a good opportunity for the use of a microcomputer graphics technique.
USING
THE P R O G R A M
E X P O F I T is a m e n u - d r i v e n , user friendly program. A f t e r l o a d i n g the program, t h e u s e r has the o p t i o n of r e a d i n g a d a t a set f r o m the d i s k or entering d a t a f r o m the k e y b o a r d . The i n i t i a l m e n u p r e s e n t e d to the u s e r is d e p i c t e d in Fig. 3.
from the analysis "RETURN" ("ENTER") the f i t t i n g r o u t i n e 5.
FITflNG ROUTINE Znter linear smoothing constant (alpha) Enter the trend smoothing constant (beta) Enter the seasonal' constant (gamma) Enter the number Of periods per cycle5 View FIT calculations Yes or No Clear plot arrays Yes or No
I. 2. 3. 4. 5. 6. 7.
Enter data from keyboard Store data on disk Load stored data V i e w / a l t e r data Fit/plot data Forecast Quit
WXICH OPTION DO YOU DESIRE?
Fig.
3.
Initial
.2 .3 .I Y N
Enter the number of periods to be used in the Inltializatlon routlne (A multiple of the number of perlods per cycle 5, I0, 15 .... )
Fig.
YOU HAVE THE FOLLOWING OPTIONS:
by p r e s s i n g the key. The m e n u for is d e p i c t e d in Fig.
5. F i t t i n g
Routine
E X P O F I T p r o v i d e s a l i s t i n g of the values of the linear, trend, and seasonal factors (denoted by SBAR, TBAR, and CBAR in the program) as t h e y are g e n e r a t e d . This listing includes the forecast produced at t h e e n d of each period. An e x a m p l e of t h i s is • p r o v i d e d in Fig. 6.
Perlod 31 32 33 34
X 35.00 33.00 37.00 40.00
SBRR 25.64 27.11 29.09 31.27
TBAR 1.58 1.56 1.64 1.75
CBAR FORECAST 1.29 26.38 1.05 38.70 1.19 40.51 1.29 32.78
Menu
The d a t a set can be l i s t e d a n d m o d i f i e d at any time. (See Fig. 4). It is a l s o p o s s i b l e to l o a d a n e w d a t a set, to save the c u r r e n t d a t a set, a n d to o b t a i n a p l o t of the d a t a b e f o r e a fit is a t t e m p t e d .
< PRESS ANY KEY TO CONTINUE >
MODIFYING DATA Fig.
OBS. No I 2 3 4 5
Xfll I0.000 23.335 25.453 29.443 21.110
CHANGE DATA? Y)es or N)o OBS. No._4___
Figure
NEW VALUE ? 19.443
4. D a t a M o d i f i c a t i o n
The user can choose to fit an e x p o n e n t i a l s m o o t h i n g m o d e l to the data by s p e c i f y i n g a c o m b i n a t i o n of v a l u e s for t h e s m o o t h i n g constants. While a s m o o t h i n g c o n s t a n t for the b a s e v a l u e is always required, the trend and seasonal components can be e x c l u d e d
6. Fit
Calculations
The use of the E X P O F I T p r o g r a m is i l l u s t r a t e d w i t h the f o l l o w i n g example. It is d e s i r e d to f i n d the c o m b i n a t i o n of smoothing constants which would y i e l d an a p p r o p r i a t e fit b e t w e e n the f o r e c a s t a n d the a c t u a l v a l u e s for the d a t a p l o t t e d in Fig. 7. This d a t a set appears to f o l l o w a r i s i n g t r e n d in conjunction with a 12-period seasonal p a t t e r n . In t h i s case a l p h a is g i v e n a s t a r t i n g v a l u e of 0.I. The t r e n d a n d s e a s o n a l s m o o t h i n g c o n s t a n t s are first d e f a u l t e d to zero. E X P O F I T a l l o w s the u s e r to e x a m i n e the c o m p u t e d v a l u e s of the base, trend, and seasonal values (denoted b y SBAR, TBAR, and CBAR in the program), and the forecast generated for e a c h p e r i o d . The c a l c u l a t i o n s are f o l l o w e d b y the p l o t t i n g routine.
Moras and B l a i r
: E x p o n e n t i a l smoothing models
375
90 80 70 60 50 40 30
e
0
|
10 Figure
7.
i
20 Data
with
e
30 trend
Since finding an acceptable forecast is a process that may require several iterations, the user has the o p t i o n of t r y i n g d i f f e r e n t values of the smoothing constants until a satisfactory match is achieved. One of the most useful features of EXPOFIT is that the most recent forecast can be compared with previous forecasts by using a m u l t i c o l o r e d plot. This feature provides a means of e s t i m a t i n g the s e n s i t i v i t y of the r e s u l t i n g forecast with respect to changes in the smoothing constants. The resulting plot can be d i s c a r d e d at any time. The superimposed plots for .the o r i g i n a l history and the basic forecast obtained with ~ = 0.i are shown in Fig. 8.
and
40 seasonal
50
patterns.
To show h o w the b a s i c model can be e n h a n c e d , s u p p o s e that t r e n d and seasonal smoothing components are included in the analysis, in this case with values of ~=0.20, 5=0.20, ~=0.25, and 12 periods per cycle. The resulting forecast is shown in Fig. 9. T h e enhanced model appears to y i e l d a closer fit than the basic model. The user may wish to try different schemes until a satisfactory match is achieved. Once an acceptable match has been ~obtained, the user can request a f o r e c a s t for k periods into the future. Since a forecast usually loses accuracy when it is u s e d to p r e d i c t l o n g - t e r m behavior, this feature should be used with caution. Continuous review of the smoothing parameters is a d v i s a b l e so that changing conditions can be accurately reflected in the model.
90 8O 70 60 5O 40 30 20
-". 10
0 Figure
8.
.
. 20
Base
model
trend
and
History
.
" 30
smoothing seasonal
40 of
data
patterns.
with
50
376
Proceeding8 of the lOth Annual Conference on Computer8 & Industrial
9O 8O 70 60 5O 40 30 2O 0
Trend & Season I
I" i
i
10 Figure
i
20 9.
Engineering
i
30
40
Enhanced smoothing t r e n d a n d seasonal
of d a t a
50
with
factors.
CONCLUDING REMARKS The p r o g r a m p r e s e n t e d in this paper provides a valuable tool in selecting the most adequate forecasting method. The usually tedious trial-anderror history matching approach is highly simplified by the speed and ease of use the m i c r o c o m p u t e r and by the v i s u a l aids p r o v i d e d by the program. E X P O F I T is e x t r e m e l y flexible in that different matching schemes can be i m p l e m e n t e d by c h a n g i n g the values of the smoothing constants. The effect of these changes can be analyzed i n d i v i d u a l l y or in c o n j u n c t i o n with previous results. By e x a m i n i n g the superimposed plots, the user can develop an intuitive sense of what the most appropriate combination of parameters should be. Thus the inevitable t r i a l - a n d - e r r o r process may become f a s t e r and less t e d i o u s . In addition to the graphical aids, EXPOFIT provides the u s e r with s t a t i s t i c a l standards commonly used to evaluate the q u a l i t y of the forecast, such as the c o r r e l a t i o n c o e f f i c i e n t and the mean absolute deviation. E X P O F I T also allows the user to try d i f f e r e n t i n i t i a l i z a t i o n schemes and analyze the effect of these a l t e r n a t i v e s on the resulting forecast. Initialization is e x t r e m e l y important when the n u m b e r of data p o i n t s is limited. The p r o g r a m takes a d v a n t a g e s of the user-friendliness of microcomputers. In addition to being a valid m e a n s to e v a l u a t e the q u a l i t y of a forecast, EXPOFIT furnishes an excellent methodology for the evaluation of the responses of a forecasting model to changes in historical data.
[i]
E. A. Silver and R. Peterson,
Decision Systems for Inventory Management and Production Planning, Second Edition, Wiley, (1985). [2]
T. E. Vollmann, W. L. Berry, D. C. Whybark, Manufacturing
and
Planning and Control Systems, Irwin, [3]
R. G. Brown, Smoothing, Forecasting and Prediction of Discrete Time Series, PrenticeHall,
[4]
(1984).
{1963).
A. C. Hax
and D. Candea,
Production and Inventory Management, Prentice Hall, [5]
(1984)
D. D. Bedworth and J. E. Bailey,
Integrated Production Control Systems. Management, Analysis, Design, Second Edition, Wiley, (1987). [6]
P. J. Harrison, Exponential Smoothing and Short-Term Sales
Forecasting, Management Science, 13, [7]
821-842
(1967).
T. D. Russell and E. E. Adam, An Empirical Evaluation of Alternative Forecasting Combinations, Management Science, 15, 1267-1276 (1987).
Moras and B l a i r
BIOGRAPMICAL
:
Exponentlal smoothing models
SMETCMES
Dr. R a f a e l G. Moras, is an a s s i s t a n t professor at the Department of Industrial Engineering, Texas Tech University. He received a BS in Industrial and Systems Engineering from Monterrey Institute of T e c h n o l o g y , Mexico, a MSE f r o m the U n i v e r s i t y of Texas at Austin, and a PhD also from UT Austin. His current research interests include applications of operations research to problems in p r o d u c t i o n and inventory systems. Dr. Moras is a member of TIMS, ORSA, and IIE. Dr. E r i c L. Blair, is an a s s o c i a t e professor at the Department of Industrial Engineering, Texas Tech University. He received a BS in Management Engineering a n d a PhD in Operations Research and Statistics from Rensselaer Polytechnic Institute. His current research interests include computer simulation, routing and logistics, and quality insurance. He is a member of TIMS, ORSA, and IIE.
377