OF IIIEEEEIIEEEEE UNCLASSIFIED AIfDAIR' ORCEI IF F L

ADI531 ARFR E IS FTC VRIHT-PATTERSON AFB OH FS1/

ALTERNATIVE METHODS OF BASE LEVEL DEMAND FORECASTING FOR ECONOM--ETC(U)DEC 75 F L GERTCHER

UNCLASSIFIED NL

AIfDAIR' ORCEI IF HI IF F

IIIEEEEIIEEEEEIEEEEEIIEEEIIE

11111_L5

ALTERNATIVE :M.ETflODS r '?

OF

BASE LEVEL DEMAND FORECAS TING

FOR

ECONOMIC ORDER QUANTITY ITEMS

A Research Effort Sponsored

by

AIR FORCE INSTII1JTE OF TECHNOLOGY,

CIVILIAN INSTITUTIONS DIVISION

and

AIR FORCE BUSINESS RESEARCH MANAGEMENT CENTER (iQ USAF)

Wright-Patterson Air Force Base, Ohio

and

THE UNIVERSITY OF ALASKA

South Central Region, Anchorage, Alaska

by

Franklin L. Gertcher

Appmed w PO82

DISCLAIMER NOTICE

THIS DOCUMENT IS BEST QUALITYPRACTICABLE. THE COPY FURNISHEDTO DTIC CONTAINED A SIGNIFICANTNUMBER OF PAGES WHICH DO NOTREPRODUCE LEGIBLY.

R e s e a-r-c! D is c 1ire r

Opinions, views or conclusions expressed by the

author in this document arc his own and are not ~

to be considered or interpreted as official ex-

pression, opinion, or policy of the Department

of the Air Force.

!?or

AcCnO):SCenV- TIG

A study of ih i- Sroec withini the ,,:vell Ljime f IMC 11 hcya I (

the ~ ~ 0 .....iit of an n niidual. L hereto-( ld:!;., C~)7-3

my -;i icerv approic in: ion to all who havc pro').' iced i~ssistLancc t his~

S pec Cia I J, C kno IIO I dgC! Mci I is e Xt en di, ed 10 C.11 t- I i 1Li1e .

Iwar t ; movcr ,Ch ie f Suppfy S s LC71s Tanalgermen t D iv j in*I'W'i

Al a.-kan Air Command and Ms. qhaan C "ibson , Computer ?ros ,rJ71Ic r, Uvr

si ly of Alaska (Anchorage). Their extensive ~o rk i n -.onve t, i 1n 1 i410.1 ;

and concepts to a FORTRAN computer program was esJseitil ro thec that a

vz l'* sphlare of th is study. Speciil ackIno%:IJ,-,wnt ~ i ;r dUtc

tc) Colonel Joseph It. Brost , Commandler , 21 Suip171V SIyJ('-Iad N,*F~ni

Air Force Base, Alaska and his staff. The ir assi stiLcL ini he data

collection phase involved over 28O man-hours Of mOlanl dati r(-;t.archi.

Til addition, 1 wish to thank Major Sanford B.' 1ozlen, R;rchAs-

sociate, Air Force Business Research 'lanagement. Center, rgt

1'Atters3on Air Force Base, Ohio, who acquired many Department o

necnse publications as background information for this study and

accomplished the necessary coordination to insure that thle rz- tilts

Of this study were reviewed by the proper agencies.

A special word of thanks is extended to my wife Judith,' for

hcr enfcourageiaelt and patience in connect ion with this arnd all of

niy other sttidi ics. Finlly, I thank my typist,* Betty Sm-.*b, for her

, o i'.er.:h c'conl. r ihutionl.

v~l~t o.f Contecnt'

Ac know ledgemn't ' .. . .. . . . . . . . . .. . . . . . . . . . . . .

List of Tables .. . . . . . . . . . . . . .. . . . . .. . . . . . .

I . lntroduct ion .. . . . . . . . . . . . . . . . . . . . ....

Prob lem Stat m n .. .. .. . .. .. .. . .. .. .. . .. .. . I

Backgroiund ...........................

)Purpose of Research .................... 3

Scope o f Research .............. ;....... 4

Obectiv . e .. . . . . . . . . . .. . . . 4

Brief Summnary of Findi[ngs ...............

Organization............................L........ .............. 7

11. Met hodology..................................................... 8

Description of Data..........................................$8

Souirce of Data........................................... 8

Time Periods for Demand Data............................ 9

Samplingv Technique.....................................

The ABC Technique......................... .............. 1I

The ABC Technique Applied............................... 11

Forecasting Methods........................... .............. 13

MovIng Averages Method..................................14

Least Squares 'Method....................................is

Single Exponential 'Smoothing Method.....................17

Double Exponential Smoothing Method.....................18

Triple Exponntial Smoothing 1,ethod..................... 19

Note .. . . . . . . . . . . . . . . . . . . . . . . . 21

AdaptivC Single Exponential Smooti-ing ........ 21

Choosing the Smoothiing Constant .............. 23

Approach .. . . . . . . . . . . . . . . . . . . . . . 24.

Method .. . . . . . . . . . . .. . . . .. . . . . . . 25

Initial JEstimaites ..................... 26

Initial Estimate: Single Exponential SmoothinE.....26

)Initial Estimate: bouble Exponential Srooth in . ........ 27

Initial Estimate: Triple Exponential Srr~othing ........ 27 -

Evaluationl Criteria for'Forecasting Methods .................29

The Mean Square Error .................. ................ 29

The Mean Forecast Error ................................. 30

The Variance........................................... 30

Statistical Inference...................................31

Final Sample Item Classifications.......................32

Standardization of Error Statistics..... ................ 33

III. Analysis of Results.............................................34

General Comparison of Forecast M ethod Performance...........35

Analysis of Forecast Method Effectiveness ............... 40,

IV. Detailed Summary......... o................4....................... 43

Experiment Summary..........................................43

Sampling Technique...................... .............. 43

Evaluation Criteria ..................................... 43

Findings ................................... 43

Error Statistic Ra41dng................................43

The Krusc~i1-Wa~llis Tvst ................ 46

Recommendi ....................... 46

Notes to the Narrative ............................o...................48

Bibliography ...........;...............................................53-

Appendixes...................................................

Appendix A: FORTRAN Compuiter Program...........................A-

Appendix B: Air' Force Tnvcntories 'lanagerient of Fxpend.ible

islipply Items........ .........................

-iv-

List of Tables

Table 1. Total Set (316 Items)......................................5

Table 2. Subset 1 (71 Items)........................................6

Table 3. Subset 2 (199 Items)....................... ................ 6

Table 4. Subset 3 (46 Items) ....................................... 6

Table 5. Absolute Average Error Distribution, SUbsor. I..............36

)Table 6. Absolute A~verage Error Distribution, Subset, 2..............37

Table 7. Absolute Average Error Distribution, Subset 3..............38

Table 8. Absolute Average Error Distribution, Total 'Se t.............39

Table 9. Results of Kruskal-Wallis Test on Forecast ErrorDistributions for the Three Top Rated ForecastingMethods................................. ................41

Table 10. Results of Kruskal-11alLis Test on Forecast ErrorDistributions for Least Squares.......................41

Table 11. Resul~s or Kruskal-tlallis Test on Forecast ErrorDistributions for Triple Exponential. Smoothing......42

Table 12t Results of Kruskal-lWallis Test on Forecast ErrorDistributions for Adaptive Single ExponentialSmoothing...............................................42

V~

1. LntrILodI(t ion

Problem Statement

The Air Force ciirrently stocks and manages 3 total of 1.48

million expendable line items in its multi-level inventory system.

The dollar investmeit in inventories of these relativel~v low cost

items is over 1.1 billion.1

Efforts have been directed toward developing and ,ppl';ing models

as approximations to reality in order to manage these large inven-

tories. These models allow managers to conceptualize the nature of

the supply environment, the inventory system, the various subsystems

within the inventory sy.9tem, and the relationships betw'een environment,

system and subsystems. An important benefit of this modeling approach

is the fact that an appropriate model can provide a means of minimizing

total inventory costs, and yet allow the system to remain responsive to

the needs of the item users.

The Air Force currently uses a modified version of the classic

Wilson lot size economic order quantity (EOQ mode] for- inventory

system management of expendable line items. The objective of this

model, designated the D062 model, is to minimize total variable costs

or ordering items and maintaining them in inventories. ,The major

itconstraints are budget restrictions and the requirement for adequate

response to the needs of the item users.

Several significant factors 'of the D062 model h,Jve not been ade-

, ' quately refined and have caused.problems in the practical application

of the model to inventories within the Air Forc supply environment.

Major problems include inadequate estimates of "eaningful cost to

order and cost to hold factors, problems of budyet constraints, and

2Inaecurate forecasts of future item demand pattrns.

Some of these problems, and others not stated, were addressed in

the development of the new A022 I"0(! model, whica the Air Force intends

to adopt as a replacement to the.D062. Many research efforts have

been initiated to make the practical application of this new model

more- refined and therefore more effective. This paper is the result

of one such research effort. The area of concern addressed by this

research efFort was the problem of inaccurate forecasts of future

item demand patterns.

Background

The need for routine forecasting methods exists at all levels in

the Air Force multi-level inventory system. Air Force base level

forecasting is particularly important since requirements at this level

drive total system stock levels. There are two primary objectives

at base level for the EOQ management model. The first is to prevent

stockouts and subsequent frustration of user needs. The second is to

manage base stock levels to minimize the sum of the inventory carrying

costs and ordering costs. Accurate forecasts of base level item demand

2i

paLterns as an input Lo the EOQ Model improves te model's capability

to meet these objectives. In tuin, base level demand patterns ire

inputs into higher echelon EO0 management svstem3, which also must

resolve problems of stockouts versus inventory costs. The key then,

to improving tothl system response in the area of stocouts versus

inventory costs. is more accurate base level forecasting of item

demand.

.

Purpoe ol Research

The present method of unweighted moving;averages to forecast base •....

level demand patterns has a tendency to over-stock inventories during

periods of declining demand and under-stock inventories during periods

of significant demand increases. This over-stocking or under-stocking

of inventories, applied to the world wide scale of the Air Force supply

environment, resul-ts in large, unnecessary inventory holding costs on

the one hand and emergency procurement (ordering) costs on the other. 3

The purpose of this research effort was to examine and compare

five methods of base level demand forecasting which can be considered

as alternatives to the present method of unweighted moving averages.

All six methods, including unteighted moving averages, were evaluated

according to relative accuracy in describing the actual base level

demand patterns of a statistically chosen stratified sample of items

managed at base level under the Air Force EOQ model.

3i

Scope of Reseairch

This study was exclusively -concerned with ependable items which

are stocked based upon D062 E0Q procedures. Expandable iLVeMS are the.se

which are "consumed in use or whichl lose their original identity during-

incorporation DIto, or attachment upon, another -issembly" J' These items,

commonly referred to as EOQ items, are designated by expendability, rc-

pa irab iIi t y, r ecov er ab iit y ca teg ory (ERRC) c od es "X 32 "and " XB3" , atmon.-

) others.__

This study was str'ictly limited to a samiple of XB2 anid XB3 itc-as

stocked by the Air Force, and present in a substantial number of base

level inventories. --The complete definitions of XB2 and XB3 ERRC coded

items are as follows:

XB2: Expendable, nonrecoverable (no repair) items with a projected

annual requirement of $10,000 or more regardless of unit price.

XB3: Expendabhle, nonrecoverable (no repair) items with a projected

annual. requirement of $10,000 or less regardless of unit price.5

Five basic forecasting methods were evaluated. These methods were:

unweighted moving averages, least squares, and the methods of sing-le,

double and triple exponential smoothing. A sixth method was generated

by modifying- the single exponential smoothing method. The modification

consisted of incorporating an adaptive exponential smoothing constant

which can be changed according to the affects of exogenous.variables.

The results obtained with this sixth method were also compared against

the results of the five basic methods.

- 4 -

-~ --

....

The valid itv of a foreca..A i:' ,. r!Ito witLa its abil ity to

perform in a real world onvirotiment. I'ierefore it was Iecided to test

the six given forecasting methods tr..ng actaal cemand experience. The

objective, then, of this ,;tudy wa,. to evaluate the given forecasting

methods using anactual sample of cconomic ordet quantity items stocked

at a base level consolidated supply activity. Ihe methods were com-

pared to determiine if -zignificant difference$ existed in the accuracics

of their respective forecasts.

Brief Summary of Findings

This summary describes the three most accurate of the six evaluated

forecasting methods over the total sample set of 316 items; and Lhree

subsets of the total set. The subs,!ts are: Subset 1, the 71 items with

relatively high level, non-erratic demand processes: Subset 2, the 199

items with relatively low level, non-erratic demand processes; and

finally, Subset 3, the 46 items with relatively erratic demand processes.

An erratic demand process means, in this study, that no discernible

constant level, linear trend or curved trend in demand was present

over the entire 15 month period for a given item.

Hean Squared Mean Forecast

Forecasting Mehod Error Error Variance

1. Single Exponential Smoothing 16.02 1.74 13.02

2. Moving Average 17.84 1.78 14.72

3. Double Exponential Smoothing 23.89 1.89 20.39

Table 1. Total Set (316 Ttems)

54

- -

Mean Squared '!van ForecastForecastn_ Mthod - Error Error Variance


2. Moving Average 76.18 5.68 44.49

3. I)ouble Exponential Smoothing 103.19 6.14 66.38

Table 2. Subset 1 (71 ILens)

Mean Squared H1ean Forecast.c.ill" _, ,Od Error Error Variance

1. Single Exponential Smoothing .36 .49 .12

2. Moving Average .37 .49 .12

3. Double Exponential. Smoothing .40 .50 .14

Table 3. Subset 2 (199 Items)

Mean Squared Mean ForecastForecastir.=Fy Method Error Error Var iance

1. Double Exponential Smoothing 3.56 1.32 1.86

2. 'loving Average 3.68 1.31 2.01


Table 4. Subset 3 (46 Items)

Single exponential smoothing provided the most accurate forecasts

for the total set of 316 items and also for component subsets i and 2, based

on rankings according to variance, mean forecast error, and mean squared

-6-

4.

error. Only on tiLhe 46' items with relatively erlItic demand processes did

the order of accuracy change. In this case, double exponential smoothing

was the most accurate iethod. Thle moving averaj;e method consistently

ranked second for the total set and subsets 1, 2 and 3.

However, a- cordlng to the Kruskal-Wallis t.st which comipared the

absolute average error distributions, no signif:-cant difference was

detected in the relative effectiveness of the six forecasting uiethods. A

)confidence interval of 93 percent was used for this test.

Organization

The remaining three chapters of this research paper are concerned

with the methodology used in the study, an analysis of results, .And a

detailed summary. Chapter I. Methodology, contains a description o

the data, a presentation of the forecasting methods and a discussion of

the txperimental design. Chapter 1111. Analysis of Results, compares

forecast method performance. Chapter IV. Detailed Summary, presents a

detailed summary of the findings, lists the limitations inherent in the 7'"

research methodology, and provides recommendations for action.

-7-:.,.

I1. Methodolog

Description of Data

All six methods of forecastJng were evaluatel according to rela-

tive refinement and accuracy in describing the actual demand patterns

of a stati;tically choseT stratified sample of items managed at base

level under the Air Force EOQ model. The methodology used in choosing

the sample is important because the results generated by. testing the

six forecasting methods apply only to the demand processes of which

the sample was representative. This section is therefore concerned

with the source of data in the sample, the techniques used in sample

selection and a description of how the sample was used in the actual

testing of the six forecasting methods.

The only source of detailed base level demand data for EOQ man-

aged items is the consolidated and daily transaction registers main-

tained at base consolidated supply activities. Such registers are

detailed records of all transactions on all active items stocked at

the base. Registers are maintained for the previous and current cal-

endar years so that historical data is available for one to two years

back depending on the current calendar month.6

Source of Data. The empirical basis for this research effort

was 15 months of demand data (I January 1973 through 31 March 1974)

for selected expendable supply items. The data source for the selec-

ted Items was the transaction registers maintained by the 21 Suply

Squadron, Elmendorf Air Force Base, Alaska.

-8-

Time Periods for Dcm,ind Data. Tie actual denand data was .aggre-

gated into 15 monthly tota'.s for each item. The first 10 monrhs of

demand history (1 January 1973 through 31 October 1973) were used as

the experience base from which to make monthly forecasts for the fol-

lowing 5 months (' November 1973 tlirough 31 March 1974).' The 10

month experience base was designated as the base Dericd.; The follow-

ing 5 months, then, was designated the forecast pariod. * The purpose

of the actual demand data in the forecast period was to use it for

error comparison in evaluating the relative accuracy of ;the given

six forecasting methods.,-

Sampling Technique. The initial sample included all XB2 and

XB3 items for three Air orce weapons/support systems. Nfhose items

which had at least one demand during the two year period 1 July 1972

through 31 June 1974 were included in the sample. The sAelection

process for the three systems was essentially random, with the

criteria that the systems be commonly deployed at many boases, and

not unique to the Alaskarn theater.

The sample items for the three chosen systems were stratified!

according the "ABC" technique. System identity was not maintained

in this stratification process. The final sample (selected from

the initial sample) was chosen to be reasonably representative of

the stratified "B" cost group of all XB2 and XB3 expendable items

u.sed in support of common Air Force weapons/support systems. A

brief explanation of the ABC technique is pertinent at this point.

-9-

The ABC Techniq uoe. lIventory management inv lves te .anage.e nt

of individual items. Unle:;s each item stocked is under 'a reasonable

degree of control, the aggregate will not be under adequate control.

A technique is needed that will isolate those iteus that have a high

total dollar invento,'y turnover per year from thcse that have a rela-

tively low total dollar turnover.

) The high dollar turnover items should be of primary concern to

the Inventory management system and should be controlled with more

precision compared to low dollar turnover items. This is si:iply an

application of Pareto's Principle of 1.faldistribution, which has been

expressed as follows: "Very often a small number of important items

dominate the results while at the other end of the line are a large

number of items whose volume is so small that they have little effect

on the results".7

Normally, input data for each item undergoing ABC analysis should

include the following:

(1) Stock Number

(2) Unit Cost

(3) Usage per time period

These data are manipulated in the following manner:

(1) Multiply usage per time period by unit cost to obtain the

dollar demand for each item.

(2) Sort the items according to dollar demand in a descending

seqitence.

- 10 -

4

(1) List this sequene. Obt:i'in2 a ctir:.lat i~ ('ollar dem-Ind fig-

tire as cacti item is added to the List.8

The listing will normally show' thit a relatively ftw itemis have

a high impact on us;age value. The "A" group is the cliassii icat ion 0.,

highest dollar demahnd items. It is usually chosen as those iterms

that miake tip 85 to 83 percent of the total cumulative d,311ar dem~and-

fig-urv, going down the list from the highest dollar dem.ArJ item

)toward 'Liie lowest. Usually the A group irncludes only ahouL 20 per-

cent of the total ntumber of items. 9 . --

The "B" group classification it, usually chosen to e 9 o9

pvrcent of the total cumillative dollar demand figure. The B group

includes those items starting at the top of the list an4 going (kowI

until. 95 to 98 percent of the cumulative dollar demand iii reachei.

The B group adds another 20 percent of the total itemns to those selec-

ted by the A group, for A total of about 40 percent of 61e total itf-ms.

The remaining, low uisage items are classified as the "C' group, which-

contains about 60 percent of the total items.fo

The ABCTechriiqe_ pLe4. The initial sample of items chosen

for this analysis included, a-, previously stated, all X32 and XB3

items required for three weapons/support systems in common Air Force

USeC. For inclusion in the initial sample, each item must have had at

least one demand during the two year period 1 July 19721through

31 June 1974. Tile three randomly chosen weapons/support systemis were: ....

the F4E aircraft, I'he C-130 aircraft and the Digital Subscriber Termi-

ital Ecqui Ipment (DSTO'V

Follow i ,, thia AlC tE niC qI. -, .the" uc, q rrd ""pit data for each it.cm

of tie initial :s:impl wer m.n i pit I at ed in LhC; f. I low i n manner

(1) The dollar demands o Lhe initial sample of Xf2 and :M items

were calculated accordinp, to:

D Pollar demand for item i (i = 1,2,3,.. ,n).

Di V1 x Pi, where),V= dem.nu volume,

year

P. = tinit price and

n = total number of items.

(2) All items were then sorted from highest to lowest value of D.

(3) After listing the sorted items, the D. for each item was added,

in sequence, cumul.atively.

nD - Total cumulative dollar demand per year.

i=l

(4) Items selected for the final sample included those starting

with the highest value of Di, moving continuously dowrn the list until

95 percent of the total cumulative dollar demand figure was reached.

The final sampl.e was therefore the B cost group of the initial

sample. This application of the ABC technique selected the most active

- 12-

X132 aizid XB'i it u.;1:, dol Iar .i--, Al l iw r 3 ,nvontcrv i ti

three rnndom Iy chki-n sy.-tem.; (hit-in,, the period I Jutly: 1972 t hr, :.;,h

31 hiune 1974. Cont inuous lemand data for the fi. 5 ,1 sample wa-i avail1-

able for thle perijod I January 1973 throuhi '31 'Izrch 1974. Th se dvc.-ar.

data for the f ina] F;ample of itcems were used to cva1.uat'': tile six fore-

casting methods considered in this- research elffart. There were 977

items in the in iti.:l .- ml.The B g;roup reduceil the final samnple

size to 316 items, aipproximately 32 percent Of thle Lot. 1

ForecaSt ing_Metllods

The objective of this recarrh vffort Was tO evaluate th- siLx

given forecastiig methods using a'..ctzilt-L sample a OQito:S;LOq L'

ait a base level consol ida ted supu . iv it"' I . Ths oect ionl of IluLS'o

part icuilar six f.orocastinog metiod., wi- I~ased onl cl~OSiII I ug mc!.I; Wh ichl

could be used rotinely and !IncxpenSively LO forecast (lemand for a

large number of rolat ively iniexpensive items b)ased ent irely Onl historiCal

demand data. The exponential. smoothing method with an adapt ive smootihing

constant was the only one of the six methods evaluated which explicitly

accounted for the effects of exogenouis variable (variablesoither than

historical demand), yet the data requirements are the same for this methnod

as for the other five methods. Other more sophisticated methods whichi

emiploy such techiniques as spectral analysis, iterat ive dynamic programming,

Bayesian analysi!;, etc., were rejected as being impractical for base

-13-

level use becaiise o f Lthe it i',hcr ltvc I o f cor'.pitt cr s to ra;*and co%,uta-

tional requirements.

rfovLing iveragfes Methodq. The process of cor~put ing *a mov'i ng aver-

age Is quite straightforward. The mathematical formul4 for calculating

the expected demand in a poriod siiply averages the demuand experienced

in the previous n periods. The average demand lorecast minimizes the

sum of the squares of the differences between the most recc-.,t n obser-

)vations and t-he expected value of the demand in the period boing

forecasted.1 it is simple and easily adapted to automiatic data

processing eq~uipmnent. Thle formula is as follows.

t n I~t-n1

where: v t is the forecast of deriand in perilod t

d. is the actual demand in period i.

n is the number of periods of actual demand which are used

to develop the forecast.

For example, let the period be one month and n 10. Then the forecast

for the first month after time t =0 is vi, an Lthe sum of the demands10

in months -9 through 0, i.e., -

0V (d_ + d- +-+ d (2)

V1 10 1"~ 10 -8 0J) 2

The effect of moving average forecasting dtpc-nds on the 1-moer. of

periods (n) ured to make the forecast. When a large number of periods

are used, the weight given to each period is relatively small, and ran-

dom Fluctuations have little effect on the forecasts. The moving

average method is more sensitive to changes in demand if n is small.

The moving average method works best when the demand generating process

is stable. If the demand generating process has a constant mean and

variance, and successive observations are uncorrelated, then the sample

mean is an unbiased estimate of the process mean. If these assumptions

are net, then the demand generating process can be considered stable.1 -

Least Squares Method. The least squares method of forecasting

assumes that the demand dt in period t can be predicted using wt =

a + bt, where a, b are determined from historical data by minimizing

F =t (d - w )2 = Z (d - a -bt)2 (3))t t: C t C

In the above equation, w t is the forecast demand for period t and d

is the actual demand for period t. Assume that a, b are to be deter-

mined by using the demands in the previous n periods. The current

time t* is a review time, and that the time period from t* to t* - T,

T being the time between reviews, will be referred to as period 0, the

period from t* - T to t*- 2T as period - I, etc. Thus the number of '

Lhe n periods to be used in determining a and b will be -(n-1). .....

-1, 0. To determine a and b, set 3F/a F/b - 0 and solve the

resulting equations for a and b.

-15-

-(n-i)aF =-2 *(d -a -bt)= 0.(4)Ia t=0 t

- (n-1)F = -2 (d -a -bt)= 0 (5)a'b t=o t

-(n-i) -(n-1)

where: * , -n(n-l); - t 2 1. n(n-l) (2n-l)t=O 2 0-

- (n-I) - (n-I)and d ;P=. td (6)

C=0 0 t

Then the solution to the'equations (4) and (5) are: .t OW!

b12 + n-iU; a =U + b(n-1)n(n-l) (n+l),L 2 n 2

so tbat w =U + bh nT

w =U + 12 2+ t = 1,2,3,.. (7)n n (n_- 1)-(+I+

Equation (7) allows computation of the forecasted demand for period t,

t 1,2,3,..., using data for the demands in periods -(n-i), ... , -i,0. 1 3

For example, let the period be one month and n=1O. Then the

forecast for the first month after time t = 0 is wl, and is:

- II. . . .

w U _ 12 P +.2:.10 ....9'0 .

-(n-1) , -n)where: U-= d ;P tdt

tnO t t=O

-16-

The least squares method, sometimes called simple linear regression,

incorporates the ability to follow a demand pat:ern which has a linear

trend. The demand process, as taken from the a'ove derivation is assumed

to be:

w a + bt

The demand In period t is a function of a, the intercept value of the

)trend line; b, the slope of the trend line and t the number of periods

between the ordinate and time t*14-

It is possible to have least squares method forecasts that are less

than zero using the given formulas. It was therefare reasonable to impose

a requirement that demand forecast must always be greater than or equal to

zero. If the forecast demand (using the formula) was less than zero for a

particular period, the forecast demand was set to zero. This procedure also

had important consequences for subsequent forecast demands, since the fore-

cast demand that %.-as set to zero was used for forecasting demands in the

following n periods.

The moving average and least squares methods for forecasting demand

both require demand data be available for n periods back. This dis-

advantage is not inherent in the exponential smoothing methods to follow.

Single Exponential Smoothing Method. Single exponential smoothing

is a forecasting method similar to moving averages in the sense that it

provides accurate forecasts for a stable demand pattern. The forecasting

formula is:

Xt adl + (I - a)Xt (8)

where: Xt the sin-le exponential smoothing forecast for: dmand in

period t,

- 17-

d the actu ji dolvifnd vxperieau:cd n period t-1,

Xt_ = the single exponeltial smoothitg forecast for demand mace

for period t-1.

the smooth ing coSLMIC, 0 < x 15

The effect'of singl,, exponential smoothinA depends on the size of

a, the smoothing constant. If a is large, more weight is given to the

most recent demand experience and the forecast is more sensicive to

fluctuations. Instead of weighting each pas.t observation equally as doas

the moving average method, the we ght given to previous observations

decreases geometrically with age. If the demand generating process has a

constant mean and variance, and successive observations are independent,

then the expectation of the single exponential forecast is equal to the

expectation of the mean of the demand generating process. If the assumptions

about the denand generating process are met, single exponential smoot!iina can

be used to produce an accurate estimate of demand. Therefore, single exponen-

tial smoothing is as accurate as the moving averages technique, while computa-

tion is simpler and storage requirements are reduced. 1 6

Double Exponential Lmoothiin~jethod. Double exponential smoothing is :.

.a forecasting method which uses a first order polynomial. Thus it incorpo-

rates the ability to follow a demand pattern which has a linear trend. The

demand generating process is assumed to be:

dt = a + bt.

The demand in period t is a function of a, b and t. Te coefficient a is

the Intercept value of the trend line; b, the first derivative of d t with

rcspect to t, is the slope of the trend line; and t is the number of periods

between the ordinate and period t*.1 7

-1,q- 1

In order LO dCVe] np th Iorec(asL YL for th s ?roce.s us ing expon.cn-

tial smoothing, coefficic-nts a and b can be est .mated using two smoothed

statist ics.

a d + (1 - a)2 (Y -dt t-l

L b + a( -d )

The one period double exponential smoothing forcast Y is then:

Y at +1)t (9)

The forecast for r periods in the future is:

,¢ = at + bT." (10)

L

Written in terms of single exponential smoothing values and .a, the

double exp.onential smoothing forecast for period t is:

Y = Xt + (I - r)Y t l1

Use of the double exponential smoothing model should be based on cer-

tain assumptions about the demand pattern. Tf the time series has a trending

av:erag2e demand rate, either increasing or decreasing, the double exponential

smoothin, method produces an accurate estimate of demand. For such a demand

pattern, the moving averages and single exponential smoothing methods la;

behind a trend, while th~e double exponential smoothing method is more

responsive in that it accounts for a trend factor. 19

Triple Exponent ial Smoothin.i ,ethod. Triple exponential smoothing

is based upon a second order polynomial time series process which ac-

counts for trend and the rate of change of trend. The assumed demand

19- t

generating pro~cess is:

dt a +bt ct2

tt

wh-re bn is the seirnt derivative of d with respect to t, vauatdcat

makes it the coefficient for the rate of chane of trend' Coef fic ients

a, b and c can be estimated by computing three smoothed .5tatistics:

at = dt-I + (1 OQ1)NZ t1 I-d t-

bt = b 3 ~2(2 .v)(Z -d )

2- tti t-

where Z is the triple exponential smoothing forecast made for periodt- 1

t-l. The forecast for period t=l. is then:

Z + +b + c (12)t t t t

The forecast made for T periods in the future is:

Z a + b T + cT 2 (13)

Written in terms of double exponential smoothing values and 04, the

triple exponential smoothing forecast for period t is:

z~ Oky + (1 - o) z (14)t t

Triple exponential smoothing is accurate when the process to be

forecasted can be adequately represented as a quadratic function of

time. Thus, if there is reAsonable justification of assuming that

the time series can be represented by a polynomial of the form

-20-

d = 1 9- )t -9 1 't 2L

then triple exponentialt smoothing provides tlIe ,,ost accurate and stable

Forecast methods. 20

NOTE: For the given three ind hi.;her orde - exoomiintial

smoothin7 metliods, Brown, in his "roof of he undjental

theorem of exponential smoothing, has demo istrated that

it is possible to e, stimate the (n-i-1) coefficients-in an nth

order polynomial. !'or the corresponding order oF exponvo-

tial smoothing% by using linea- comibinations of the first (n-:-1)• .. 4 ;. -.-

orders of exponential smoothing. It shoulM be noted at

this point that Brown's proof can be used to derive

formulas (8), (11), and (14).21

Adaptive Sinyle Exnonential Smoothing. t The sinc, le exonential

smoothing forecasting equation is:

Xt = dt_ 1 + (.- a) Xt (15)

Let Xt equal the adaptive single exponential! smoothingl forecast for period

t, and let a* be an adaptive smoothing constant. The adaptive single

exponential smoothing formula is therefore:

X* =*dt_1 + (1 - a*) Xt 1 (16)

The single exponential smoothing method provides good forecasts

w!icn the demand process has a constant mean and variance. However,

- 21-

actual demand is often a;ub~ect to sudden incremental changes due to

variables exogenous to the model. For example, the number of systems

for which an item is used may be suddenly increased ot decreased. Under

these conditions, as the level of demand changes, the single e:.onential

smoothing formula will change the forecasts over time, moving towards the

new demand curve at a rate dependent on the value of a, the smoothing

constant. The adaptive aspects of this adaptive single exponential

smoothing method were b.il.t upon'this feature. 2 2

The basic philosophy was to change a* to a value of one when sudden

shifts in the actual dempand level, d, were detected. This has the

advantage of changing the estimates of the mean demand, X*' almost

immediately to the range of the new demand mean. The procedure to

accomplish this was based on the observation of actual: demand '.;lues

outside an acceptable range around the current forecast.

It is assumed for the purposes of this model that the random

fluctuations about the basic demand level, d, were normally distributed

with a known, constant standard deviation. In such a case, it is

natural to use a set of controls on X* which permit a rapid response tot

abrupt changes in demand patterns. There is, however, a difference

between the problem here and the apparent analogy with the classical

quality control case of, controlling a process by detecting points out-

side control limits set around the desired Process mean. The difference

is brought about because the initial demand Forecast is likely to be

- 22 -

different from tIe Lrue mv an demand by an amount dependt on t ., ;oth-

Ing constant and the demand sequence. This possible error means that a

single actual demand, dt, may differ sufficiently frcra the current fore-

cast to lead one to suspect that the basic demand has shifted when, in

fact, it has not.

Because of this complication, two criteria were developed for

concluding that a new demand level had been, detected and changing the

).smoothing constant is necessary. The first of these criteria is ap-

plied in the case where a single demand is discavered -outside a

range of four standard deviations around the current rjean of actual

demand. The second criteria is applied when two successive demands

differ from the current estimate by more than 1.2 standard deviations

and both of thesc points are either above or below th mean actual

demand.

The second consideration in the development of the adaptive

aspects of the model was the choice of the values of the smoothing

constant after the detection of a shift in the basic demand level.

The model uses a 3-stage change of the smoothing constant. The

constant is set equal to I for the first period after:a change in the

basic level of demand has been detected, and to .8 Cor the second period

and .5 for the third period. In the fourth period the value of a is

returned to its "normal level.!F.

Choosin -the Smoothing Constant

-23 -

_Approach. The smoothing constant, O, is tisel in each of the

exponential smoothing forecasting.m.erhods. Accorling tcr Brown, the

value of Ot is usually chosen betueen .01 and 0.3. Values between

0.3 and 1 are normally not used, since these values cause the formula

to 'be too sensitiye to random, non-representative fluctuations in the

demand generating prccess. For the exponential smoothing methods

evaluated in this study, values of o0 between .05 and .5, in increments

) of .05, were considered. The value of ok which minimized the standard

deviation of forecasting errors for the single exponential method was

chosen as the optimum o.

The purpose of the smoothing constant is consistent in all of

the exponential smoothing methods. This purpose is to weight the past

demands used in the forecasting formula in a single, prescribed

manner. The ot chosen should strike a balance between responsiveness

to demand process changes (value of A approaching 0.5) and damping of

random fluctuations in the demand process (value of Ot approaching .05).

It should also he noted that in this study, the same set of demand data

was operated on by all forecasting methods. Considering the purpose

of otand the use of the same demand data, it was reasonably assumed

that the optimum value of cK for single exponential smoothing approxi-

mates the optimum value of A for double and triple exponential smoothing.

The Ok that minimized the standard deviation of forecast errors for the

single exponential smoothing method was therefore also used as the

optimum A for the double and triple exponential smoothing methods.

--24"

This optimum 0 was also ustd as the "normal" value of o for Lae adap-

tive triple exponential smoothing method.

Hfethod. Specifically, the optimum O from the set .05 < a 0.5

in increments of .05, was selected in the following manner:

I

(1) Xt = Ad 1t- + (1 - 00 X tl, (17)

where X = the single exponential smoothing forec3st for period t.

X single exponential smoothing forecast for periodit-i, exceptt-l 0

Xt_ X where X0 d.It-l 00-l~i=-9i

(2) Find Xt for each item for each, value of 4, .05 < a < 0.5.

There are 10 values of X , one for each X, for each item per month.t

This was repeated for all 316 items for each month of the forecast

period, 1 November 1973 through 31 March 1974.

(3) The differences between actual demands d and forecasti

demands X for each month of the forecast period were found for eacht

)item, for each M.

Let c(. for j = 1,2,3,...,10 and1

e 5 IX - d i, (j 1,2,3,...,l0) (18)IN, 5 i=t=l t dj

Repeat for each item.

(4) The mean for each e was then found.

j (e) n = 312 items. Repeat for each A . (19)

(5) The standard deviation for each e was then calculated.

- 25 -

s standard deviation (e, A4) 2 (20)

n-

Repeat for each 0(.

The value of A which produced the smallest tandarL deviation,

s, was chosen as the optimum A. That value of O ,was then'used in the

calculations for the given exponential smoothing rethods.

Initial Estimates

The first 10 months of demand history were used as the experience

base from which to make monthly forecasts for the following five months.

The smoothing constant C.was selected based on the five month forecast-

ing period. The exponential smoothing methods also require initial

estimates for the coefficients X, Yt-l and Z when :t-1 - 0.t t-1 t-l

These initial estimates can severly bias method performance over a23

small number of periods. Since the weight assigned to an estimate

decreases geometrically with age, the effect of a biased initial

estimate eventually becomes insignificant. However, in this study,

the forecasting period included only five months. Reasonable care

then, had to be taken in determining the values of the initial

coefficients.

Initial Estimate: Single Exponential Smoothing. A'common and

normally sufficient approach to providing the initial estimate of

coefficient X for single exponential smoothing is to take a simple0

average of a selected number of data points.2 4 In this study therefore,

-26-

the initial coefficient X0 was determined by taking an average of the

actual demand over the 10 mon-th base period.

0

X d (2.1)

Initial Estimate: Double Exponential Smoothing. The double

exponential smoothing method requires initial estimates for both the

slope and the intercept coefficients a and b:

Yt = a + bt (22)

The method used to compute Y was a simple linear least squares0

regression based on the 10 data points of the base period. The

least squares estimates of a and b served as the components of YO

for the forecast Y The remainder of the 5 months of the fore-1

casting period were forecast using the standard double exponential

smoothing formula.

Initial Estimate: Triple Exponential Smoothing The triple

exponential smoothing method requires initial estimates-for the

intercept, slope and rate of change of slop of the demand process

equation.2 5 Consider the assumed demand process:

dt M a + bt + ct2 (23)

taOver the 10 data points of the base period,

d (demand)

a -9 -8 -7 -6 -5 -4 -3 -2 -1 0 t (time)

Demand Versus Time Distribution

-27 -

The value of a is the d axis intercept of d , and can be. estimatedt

by:

a d (24)

Thelvalue of b is the limit as At approaches zero of --. Using data -:'

At 4i

points at t -9,-7,-6,-4,=3 and -1;

) K 7) +(d-6 ) + d

zii3-

which reduces to:

b (d -d +d -d +d -d ) (25)6 9 - -6 -4 -3 -1

The value of c is equal to the rate of change of b with respect

to t. Consider the rate of change of b using two data points of b;

1 I - d d d d 4 ) -(d 3 -dS21 ;2 3- L- 2-

which reduces to:

1c - 72 (d -d -d +d ) (26)

2 9 -7 -3 -i

The following equation was taken from Brown's book, Smoothing,

Forecasting and Prediction of Discrete Time Series, and is the triple

exponential smoothing demand process quadratic formula in terms of a,

b, c and CC.26 In this form, the demand process equation can be used

to estimate Zt. For Z0 then;0

28-

Z = a - 3(1 - ) + 3(1 - o)(4 -3) (27)

The range of values taken on by Z0 must be 1 mired to positive

demands. It is also reasonable to impose an uppe:" limit of 2a, so

that:

0< Z0 < 2a

When Z calculated by formulas (24), (25), (26) azid (27) is negative,0

let Z0 = 0. When Z calculated by (24), (25) (26) and (27) is > 2a,

let Z 2a.0

The value of Zt_ I at t-l = 0 was calculated in the manner pre-

scribed above. The values of Z for subsequent periods of course,t

were calculated using the standard triple exponential smoothing formula.

Evaluation Criteria For Forecasting Methods

A comparative evaluation of the accuracy of the six forecasting

techniques was performed in terms of the forecast error each produced.

Three statistical measures of error were selected: the mean square

error, mean forecast error and variance of error.

The Mean Square Error. The mean square error (MSE) is a measure

of forecast accuracy which emphasizes large errors. 2 7 The mean square

error for each item and forecasting technique was computed by sumning

the squared forecast error for each'of the 5 months and dividing by the

number of months forecast. The equation for the MSE is deveclped from

the definition of forecast error:

5

- 29 -

I

where e the absolute average forccast error cver the 5 ,.aonth fo -c.sti;

period

R wV W Y Z , or X depending on the forecast metho4ta t

used.

d.t = the aptual demand in period t. -

hen the mean square error is: , -

MS (e (R - d.)The mean square error for each itoe was summed and divided by th, sample

size to obtain an average mean square error. The forecasting technique

with the smallest average mean square error was termed the most accurate

in that it mfnimi.ed the number of large errors.23

The Mean Forecast Error. This statistic was computed by averaging

the absolute value of the forecast errors fqr each item by forecast model.

The equation for the mean forecast error is:t

n C-1 i n t4 I ti

The average mean forecast error was then computed for each forecasting

technique by summing the 'IFE for all items and dividing by the number oi,

items. The preferred model was that for which the average MFE was closest

to zero.

The Variance. The third comparative statistic was the variance of

forecast error (VAR). A high variance indicates an unstable or erratic

eatimator, while a small variance indicates a tcoi-s:cnt est,,azor.

- 30 -

The var Lance of forecast error was computed for each item for each-

forecast model. The eqtiation for the varianc a% forecast error is:

R (e -MFE)n1tal t

iS

The variances for the sar,-ple were then averaged for each model. The

model with the smallest average variance was co~isidered the Most stable

estimator.

Statistical Inference. At this point, it is pertinent to discuss

statistical inferences concerning-the error statistics of the six

evaluated forecasting methods. A nonpararietric test was necessary to

evaluate the error statistics since no specific assumptions could be

made about the distribution of the error population. ,onparametric tests

are not concerned with parameters of a specific type of distribution, and

no assumptions are made about the populations sanpled.

The nonpirametric Kruskal-4allis test (I test) was used to determine

if there was a significant difference in the effectiveness of the six

forecasting methods evaluated in this study. The 11 test evaluates the

null hypothesis that k independent random sample distributions come from

identical popula' ions against the alternative that the means of the

populations are not all equal.30

In this case, the absolute average error distributions generated by

each forecasting method were considered the independent sample distributions.

Within each subset, the error distributions were compared to determine if

they did or did not come from identical populations. If the pertinent

- 31 -

error d litribut ions were f roun identLi Ca] popuilat ;Luns, then there were

no significant differences in the effectiveness Of 0hC resp.c~ivt; fore±-

Casting inetthod:s. On the other hand, if they wec not from identical

-iult ions, then Lthe re were s ignif ican t dif ferences in eff cc tiv ness

The combined bbsolute average errors from the forecasting methods

being compared were ranked according to size. R.i was the sum of the ranks

assigned to the uiobservations in the ith method and n n, + 1a2 +...

) + 11n The te,;t viaf based- on the statistic2

11= 12R.- -3 (n+l)n(n+l) n

If the null hypothesis was true, the sampling distribution of this statistic

could be approximated closely with a chi-square distribution with k-i

deorees of freedom. Consequently, the null hypothesis at the level of

2significance Z could be rejected if Hi exceedsXA .for k-I. degrees of freedon.

AnX?05 was used to give a confidence interval of 95 percent.

Final Sample Ttem Classif~ications. From an examination of the toal

demand data and the error statistics for each forecasting mtcthod, a relation-

ship was noted between the demand characteristics of an ft.l .-nd the

magnitude of its absolute average forecast error. items with high .'10i-and

rates tended to have larger error statistics than did items with lo'u

demand rates. Consequently, the former played a more significant role

In the determination of the best forecast model. In order to equalize

the effect that items had on this determination, Lotal ofa - ite7_3

w.as divided into three subsets based on the deriand process chara-cteristics

of each~ item. The six demand forecasting miethods ovaluated were therefor,.

*1 -32.-

applied to the foltowin 4 set and subsets of the 31l item final sazpj.."

(1) The total .;et of 316 items.

(2) The subset which included the 71 items with relatively hig,-,

level, non-erratic demand processes. Non-erratic means, in this study,

a relativel.y constant demand process, or a demand with a discernible

t~end (linear or curved)'.:.Tr

(3) The subset which included the 199 items with relatively low-

level, non-erratic demand processes.

(4) The subset which included the.46 items with erratic deaad

processes (no discernible constant level or trend, linear or curved).

Standardization of Error Statistics. Consideration was given to

standardizing the absolute average error statistics used as a basis for

evaluating the six forecasting methods. For example, the absolute average

error for each item could have been divided by the average monthly demand

rate for that item to produce a standardized statistic independent of

demand level. However, the absolute average error for each item would

have been divided by the same average monthly demand rate for each fore-

casting method. The standardization process would not change the relative

ranking of the forecasting methods. Standardization of the absolute

average error statistics was therefore not used.

- 33 -

of Results ~- 3~*,~.HI. Analysis -.

4'

.3

1~

General Comp.iison of Foreca.:t

Method Performance

35

Freq enc25

20 Subset 1 (71 Items)

Absolute Average Error Distributions

15

10

5

Average Error

0 .5 1.0 1.5 2.0 2.5 3.0 3.5. 4.0 4.5

Mlean Squared Mean ForecastMethod Error Error Variance

- Single Exponential Smoothing 67.99 5.51 38.15

Moving Average 76.18 4.68 44.49

- Double Exponential Smoothing 103.19 6.14 66.38

Least Squares 108.92 6.24 71.02

- Triple Exponential Smoothing 124.67 7.18 74.21

Adaptive Single Exponential 120.78 6.22 83.28

Fre uenc

125

Subset 2 (199 Items)

100 Absolute Average Error Distributions.

75

50

25

Avera e Ercr0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Mean Squared Mean ForecastMethod Error Error Variance

Single Exponential Smoothing 0.36 0.49 0.12

Moving Average 0.37 0.49 0.12

Double Exponential Smoothing 0.40 0.50 0.14

Triple Exponential Smoothing 0.41 0.49 0.17

Least Squares 0.47 0.53 0.20

Adaptive Single Exponential 0.52 0.56 0.20

Freg enc •

Subset 3 (46 Items)

25

Absolute Average Error Distributions .

10.-.:-

5

AeaeError

0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Mean Squared Mean ForecastMethod Error Error Variance

Double Exponential Smoothing 3.56 1.32 1.86

- Moving Average 3.68 1.31: 2.01

- Single Exponential Smoothing 3.81 1.35 2.O4

-Adaptive Single Exponential 4.51 1.51 2.28

-. Least Squares 4.46 1.32 2.79

-Triple Exponential Smoothing 5.52 1.46 3.47

Fre uenc9-.

125

Total Set (316 Items)

100Absolute Average Error Distributions

75

50

25

0 ~ 5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Mean Squares Mean ForecastMethod Error Error Variance

Single Exponential Smoothing 16.02 1.74 13.02

- Moving Average 17.84 1.78 14.72

., Double Exponential Smoothing 23.89 1.89 20.39

-Least Squares 25.36 1.92 21.72

- Adaptive Single ExDonential 28.05 1.97 24,24

Triple Exponential Smoothing 29.00 2.13 24.55

Analy-siS of Forecast !4etholc

Effectiveness

-40-

0

~ .41

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C~wc V

44 $4 -4

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W- P-4 ai

tojU-dy

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ciac

i-i.' H 'AC4 t 0

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IV. Detailed Summary

Experiment Summary

Sampling Techniquqe. There are many possib ..e ways of seleccing a

sample of EOQ iiems. One method is by stock nuiber classification.

Other methods are demand level and cost. In this study, the scrati-

fled base level X32 and XB3 item sample (B cost group) was selectcd

from three r~ndomly chosen systems. Within the toal sample set, iteas

were grouped according to demand level (subsets 1, 2 and 3). These

important facts should be kept in mind when drawing conclusions from

the results of the analysis.

Evaluation Criteria. The forecasting methods were ranked according

to error statistics. Variance, mean forecast error and mean squared

error were the error statistics used. The smaller the error statistic,

the more accurate the forecasting method. The Kruskal-Wallis test was

used to determine whether there was significant differences in the

relative effectiveness of the six forecasting methods.

Findings

Error Statistic Ranking. Single exponential smoothing proved to be

the most accurate alternative to the present method of moving average

forecasting. Double exponential smoothing also ranked in the top three

for the total sample set and for subsets 1, 2 and 3. L ,

triple exponential smoothing and adaptive single exponeat-a. zohi.'g

ranked fourth, fifth and sixth, respectively. The followin, ?z,:gra~hs

discuss the reasons for their relatively poor showings.

-43-

The lea!t squares 7, Lhod tended to respond to au,!d,,r., -r-e,

non-representative demands by excessive changes in the slope of tihe

trend line. In some cascs, this change of slope resulted in forecasts

of negative demands for the following period or periods. This type,

of erroi was mid'1gated to some extent by settin,, all negative least

squares forecasts to zero. On the other hand, double exponential

smoothing, which also assumes a linear trend in demand, gave reiatively

more weight to demand history with subsequently less drastic changes

in trend. The non-negative demand constraint was not necessary for

double exponential smoothing.

The two primary causes for the relatively poor showing of triple

exponential smoothing were the absence of demand processes uhich could

be accurately approximated by a second order polynomial, and the poor

results obtained with the formula for initial conditions (Z0 ). None

of the 316 items in the sample had a demand pattern with a consistent

rate of change of slope. The formula used to calculate initial condi-

tions, though mathematically correct, was therefore not representative

of the actual demand processes. Starting with relatively inaccurate

initial conditions, the errors tended to compound during the forecast

period.

The results obtained with the adaptive single exponential smooching

were also disappointing. The major cause of error relative to standard

single exponential smoothing was the first criteria for detactiag a

change in demand level; i.e., a single demand detecae4' outsfc zur

standard deviations of the mean demand. When such a cc~.iaa. was detec-eu,

-44 -

the smoothing con.tant .Jis changed to one. This crLte-i.:2d :he sz' Se-

quent smoothing constanc change caused the fore:ast to ch.avk,;e excessively

in response to abrupt, nn-representative demands. The. second criteria;

i.e., two successive demands differing from the mean demand by more '

II1.2 standard devIations, both demandsbeing abova or below the mean,

appeared to change the smoothing constant at appropriate times for the

given sample.

Three obvious approaches to improving the. accuracy of adaptive

single exponential smoothing are open. The first would be to drop the

first criteria for changing the smoothing constant, retaining only the

second criteria. The second approach would be to change the maximum

smoothing constant to some value less than one. The third approach

would be a combination of the first two approaches. However, these

possible improvements were not attempted in this study and are lei as

subjects for further research.

Finally, a few words concerning the choice of smoothirg constants

for double and triple exponential smoothing are In order. It was

assumed that the optimum smoothing constant for single exponential

smoothing would also be optimum for double and triple exponential

smoothing. No proof was offered for this assumption. It would be of

interest to compare the accuracy of double and triple exponential

smoothing forecasts using smoothing constant (C) values in Increments

of .05 between .05 < a < .5. This comparison was not attemt- for

this study, and is also left as a subject for pertinent fuure research.

- 45 -

'rile Kruskal -Wal is &'st .[his test indica:ed that tic popul.,iCons

of absolute average forerast errors ire nearly identical for the six

forecast methods. 11ased on this test, there were no significant dif-

ferences In the relative effectiveonss of the s,x methods. However, as

previously no ,d, there is substantial evidence that improvement is

required in t!o. formulition of least squares, triple exponential

,moothinT and adaptive single exponential smoot.;ing forecasts. It

would seem appropriate :then, to withhold final judgement on these three

forecasting methods until the suggested improvements are made and

tested.

Recommendations

The purpose of this research effort was to examine and compare

five methods of base level demand fore'a-ting which can-be considered

as alternatives to the present method of unweighted moving averages.

All six methods, including unweighted moving averages, were evaluated

according to relative accuracy in describing the actual base level

demand patterns of a statistically chosen, stratified sample of XB2

and XB3 EOQ items.

There is little doubt that the single exporential smoothing maethod

of forecasting item demand is a viable alternative to the present

method of unweighted moving averages for items represented by the

sample used in this study. Double exponential smoothi-v, also appears

viable.

- 46-

Research using the given for:cst ing methods should cc.rrainlv

Continue. Pccorrmjend that subsequent research eiforts use e sumples

gathered by differet methodolgies at base levl. In thisq way, a

broad base of item samples, wholly representati"e of arl EOO items

Air Force wide, can be established. Given that the results o- future

research efforts bear out the findings of this ;tudy, serious consider-

ation ought to be given to single and double exponential smoothing

methods as alternatives.to the present method of forecasring deraands.

Single and double exponential smoothing have the well documented,

inherent advantages of smaller data storage requirements, greater

computational ease, and greater flexibility in adapting to demand

pattern trends compared to unweighted moving averages.

Further research, a- indicated under the Findings Section of this

chapter, is necessary to properly evaluate all pertinent improvements

concerning least squares, triple exponential smoothing and adaptive

single exponential smoothing. Relatively simple adaptations to the

FORTRAN program used in this research effort can be made to analyze the

suggested improvements. Recommend these adaptions be made and the

program used to evaluate the given sample and other base level sarmples

selected by different methodologies.

- 47 -

Notes to t!ie Narrative

1. Coile, James T., Captain, USAF, and Dickens, Denns U., Captain,

USAF, HisLor" .tnd v,;.luat ion of the Air Forzce Depot LOvel EOQ

Inventory Model A Thesis, presented to the Faculty of the School

of Systems and Logistics, Air Force Institute of Technology, Air

University, Wright-Patterson Air Force Base, Ohio" January 1974.

pp. 1. McManus, O(qen, CS-12, Maragement and Procedures Branch,

21 Supply Squadron, Elmendorf Air Force Base, Alaska, Several

Personal Interviow, "etween 2 July and 1 December:1974.

2. U.S. Department of 'the Air Force, Selected Areas for Business

Research, Air Force Business Research Management Center, Wright-

Patterson Air Force Base, Ohio, 'fay 1974, pp. 7. :Brost, Joseph R.,

Colonel, USAF, Commander, 21 Supply Squadron, Elmendorf Air Force

Base, Alaska, Numerous Personal Interviews Between 2 July and

1 December 1974. McManus, Owen, S-12, Chief, Management and

Procedures Branch, 21 Supply Squadron, Elmendorf Air Force Base,

Alaska, Several Personal Interviews Between 2 July and 1 December

1974.

3. U.S. Department of the Air Force, Selected Areas for Business

Research, Air Force Business Research Management Center, Wright-

Patterson Air Force Base, Ohio, May 1974, pp. 7.

4. Fischer, Donald C., Major, USAF, and Gibson, Paul S., Captain,

USAF, The A.licationof Exponential Smoothing to Forecasti.,

Demand for Economic Order OuntitLy tems, A Thesis, presented to

- 48 -

the Faculty of the School of SysLers and Lo:.istics, Air Force

Institute of Technology, Air University, Wv..ght-Patterson Air

Force Base, Ohio, 28 Januairy 1972, pp. 8.

5. U. S. Department of the Air Force, R equiremnts Procedure. for

Economic Ord6.rQuantitv (EO) _Items, AFLCH 37-6, Wright-Patterson.

Air Force Base, Ohio: leadquarters Air For :e Logistics Co._-n and,

14 February 1973, pp. 1-8.

6. *Uctlanus, Owhen, GS-12, Chief, .anagement and Procedures Branch,

21 Supply Squadron, Elmendorl'Air Force Base, Alaska, Several

Personal Interviews Between 2 July and 1 December 1974.

7. Killeen, Louis M., Techniques of Inventory Management, American

Management Association, Inc., 1969, pp. 20.

8. Ibid., pp. 20, 21.

9. Ibid., pp. 22.

10. Ibid., pp. 22-25.

11. Brown, Robert Goodell, Smoothing, Forecasting and Prediction of

Discrete Time Series, Prentice-Hall, Inc., Englewood Cliffs, New

Jersey, 1963, pp. 98, 99.

12. Fischer, Donald C., Major, USAF, and Gibson, Paul S., Captain, USAF,

The Application of Exponential Smoothing to Forecasting Demand for

Economic Order Quantity Items, A Thesis, presented to the Faculty

of the School of Systems and Logistics, Air Force Institute of

Technology, Air University, Wright-Patterson Air Force Base, Ohio,

28 January 1972, pp. 17.

- 49 -

13. Iradely, C., and hictn, T..i. , A I v;is of -I ontory

Prentice-|lall, Inc., Englewood Cliffs, 'New Jersey, 1963, pp. 415-417.

14. Fischer, Donald C., V'ajor, USAF, and Gibson, Paul S., Captain, USAF,

Exonential Smoothing to Forecasting Demand or

Economic Order Quant tItems, A Thesis, pr.sented to the Faculty


Technology, Air University, Wright-Patterson Air Force 3a.e, O.io,

28 January 1972, pp. 18.


Discrete Time Series, Prentice-Hall, Inc., Englewoqd Cliffs, New

Jersey, 1963., pp. 102. Fischer, Donald C., Major. USAF, and

Gibson, Paul S., Captain, USAF, The Application of:Exponential

Smoothing for Forecasting Demand for Economic Order Ou-antitv Items,

A Thesis, presented to the F~iculty of the School o Systems and

Logistics, Air Force Institute of Technology, Air University,

Wright-Patterson Air Force Base, Ohio, 28 January 1972, pp. 18-19.

16. Brown, Robert Goodell, Smoothing, Forecasting and 2redlction of

Discrete Time Series, Prentice-liall, Inc., Englewood Cliffs, New

Jersey, 1963, pp. 100-102. Fischer, Donald C., Major, USAF, and

Gibson, Paul S., Captain, USAF, The Aplication of Exponential

Smoothing to Forecasting Demand for Economic Order'Quantity Items,

A Thesis, presented to the Faculty of the School of Systems and

Logistics, Air Force Institute of Technology, Air University,

Wright-Patterson Air Force Basb, Ohio, 2g January 1972, -,p. 17, :. .

- 50 -

.. ... . ... . . I I Il -I I i ii l _ -

17. Brown, Robert Coodell, Smoothinr,, IVorecast i:ng and Predliction of

Discrete Time Series, Prentice-11all, Inc., ".nglewood Cliffs, New

Jersey, 1963, pp. 128.

18. Ibid., pp. 128-144.


The Application of Exponential Smoothing to Forecasting Demand for

Economic Order OuantiLv Items, A Thesis, presente to the Faculty


Technology, Air University, 'Wright-Patterson Air Force Base, Ohio,

28 January 1972, pp. 18, 19.

20. Ibid., pp. 20.


Discrete Time Series, Prentice-Hall, Inc., Englewood Cliffs, New

Jersey, 1963., pp. 132-144.

22. IWbybark, D. Clay, Testing an Adaptive Inventory Control Model, A

0 Working Paper, Division of Research, Graduate School of Business

Administration, Harvard University, Soldiers Field, Boston,

Massachusetts, 1970, pp. 4. Martin, Merle, Associate Professor,

Division of Stusiness, Economics and Public Administration, University

Of Alaska, Anchorage, Several Personal Interviews netween I October

and 15 December 1974.


The Application of Exponential Smoothing to ForecaszS i c.? a-d for

Economic Order Quantity rtems, A Thesis, presented to the Faculty

of the School of Systems and Logistics, Air Force Tnstitute of

- 51-

Technology, Air Univi.rsity, V./right-Patter- o; Air Forca Base, Ohio,

28 January 1972, pp. 21.

24. Brown, Robert Coodelt , SmoothinEg, Foreca.ti-g and Prediction of

Discrete 'rime Series, Prentice-flall, Inc., -Tnglewood Cliffs, New

Jersey, 1963, pp. 102.


The Appication of Exponential Smoothing to Forecasting Domand for

0 F-onemic Order Qtanti r_ Items, A Thesis, presented to the Faculty


Technology, Air University, Wrigbt-Patterson Air Force Base, Ohio,

28 January 1972., pp. 22.

26. Brown, Robert Goodell, Smoothing, Forecasting and Pre-iction of

Discrete Time Series, Prentice-1lall, Inc., Englewood Cliffs, Ne

Jersey, 1963., pp. 136.


.The _plicat ion f Exponential SmootY-in Lt r Forecasting Demand for

Economic Order Quantity_ Items, A Thesis, presented to the Faculty

of the School of Systems and Logistics, Aiz Force Institute of


28 January 1972, pp. 24.

28. Tbid., pp. 25.

29. Ibid., pp. 26, 27.

30. Freund, John E., Hodern Elementary Statistics, Prentice-Ball, Inc.,

Englewood Cliffs, New Jersey, 1973, pp. 376-377.

- 52-

1. Brost, Jos eph R., Colonel, USAV, Commander, 21 gI!pply Squadron,

Elmendorf Air Force Base, Alaska, Numberous Personal Interviews

Between 2 July and 1 December 1974.

2. Brown, Rober't -oodell, Smoothin _1_orcastii and Prediction of

Discrete Time Series, Prentice-H1all, Inc., i:nglewood Cliffs, New

Jersey, 1963.

3. Coile, James T., Captain, USAF, and Dickens, Dennis D., Captain,

USAF, History and Evaluation of the Air Force Depot Level EOQ

Inventory Model, A Thesis, presented to the Faculty of the School

of Systems and Logistics, Air Force Institute of Technology, Air

University, Wright-Patterson Air Force Base, Ohio, J .nuary 1974.

4. Fischer, Donald C., Major, USAF, and Gibson, Paul S., Captain,

USAF, The Application of Exponential Smoothing to Forecasting Derand

for Economic Order Quantity Items, A Thesis, presented to the Faculty

of the School. of Systems and Logistics, Air Force Institute of


28 January 1.972.

5. Freund, John E., Modern Elementary Statistics, Prentice-Hall, Inc.,

Englewood Cliffs, New Jersey, 1973.

6. Hladely, G., and [Chitin, T. M., Analysis of Inventory Systems,

Prentice-liall, Inc., Englewood Cliffs, New Jersey, 1963.

7. Killeen, Louis M., Techniques of Inventory :anagement, .U-erican

Management Association, Inc., 1969.

- 53-

S. Martin, . vrlc, A. sociate Professor, Divisioni 'f uT.iness, Economics

and Public Administr; ion, University of Aliska, Anchorage, Several

Personal Interviews Eetween I October and 13 December 1974.

9. McManus, Owen, GS-12, Chief, Manapement and Procedures Branch,

21Su pply Squadron, Elmendorf Air Force Bas , Alaska, Several

Personal Interviews Between 2 July and I De.ember 1974.

10. U. S. Department of the Air Force, Renuirements Procedures _or

Economic Order Quantity (EOQ) Items, AFLCM 57-6, Wright-Pacterson

Air Force Base, Ohio: Headquarters Air Force Logistics Command,

14 February 1973.

11. U. S. Department of the Air Force, Selected Areas For Business

Research, Air rorce Business Research Management Center, Wright-

Patterson Air Force Base, Ohio, 'May 1974.

12. Whybark, D. Clay, Tetin_ an Adaptive Inventory Control Model, A

Working Paper, Division of Research, Graduate School of Business

Administration, Harvard University, Soldiers Field, Boston,

Massachusetts, 1970.

- 54 -

Append ixes

-55-

Append ix A

FOAT RAN Comput er Program

AA

IOEN4T MAIN1tvI r 5 -N~?- gu " 7-,N'rZ -E~

FILE 6 OUTPUT # UNIT = PRINTER__________-"I C10=Q~I, -Li ~ .nRE-~PRNtHREOR-L-9FILE It OSKrL2,UNIT= I SKWORKF1I.ERANONRECCRO2300

-4- MENSIN- NSN(4t),O8SVO( 15),CALC(20).PE91CD(14 SOG-M1

ALPHA NSN

INN 25

C

CALC(N) 2ZERO

00 3 M IAn

51 FORHAT(2Xp4A4p15F1.0)itG UX.T-z- KQUbNT .L-. - _______________ __

52 FORMAT(4A,,I5F4-0p20r6.2)

4 WRITE(VnUT*60)Ad -5 FRHA.T(1R1,#0xp31HiS.fl0THING C0oNf-TAN-T- CAt.CULATIONSs/IA2.X2414Uk..t-ONA-

IL. STOCK N4UMBERIX,514M.0O5,5X,5HA. 10,5X,5HA=. 1p5Xp5HA:.20,5> pSHA2

2w25XHA=-930t,5X' 5liAe-35p5X,-5HA.*4O'5XASHAz..45s.5X,.SHAZ ,.-.5AAL)-----

00 5 KIlptOALP-ALR*4.5CONSNT(K)zALP

5 CONTINUE

N1l

00 6 1:1.10

A-1

VlaV/1O.O

00 8 KsIIO

DIFF=O.O

tMUJ+9

Xlz(CONSNT(K).O8SVO( I))4((1.O'CflNSNT(K))* VI)

: 1FziF AGS X 1-O8SVO(L) )

CALC(K)=OIFr/5.o~ -- _______ ____________ ____________

8 CONTINUE

-- O-S T --1 = 0STX(I)=O.O

STZ(1)0.0O______Si2A(L)-Q0-_____________ ________

47 CONTINUE

61 FORMAT(IH ,3XpAQ,3H 00 P3AqpXpr6.2,'lCFlO.2))

~~5 A FORFATC5or6.2) - _______________________

MARK=MARK~tTFQ'HARK- '(YO'NI) ir a 2 1CO Uft T:KOUNT

HOLOCCI )ZERO9--OR C )=PE IOD I -_ __ __ __ __ __ __ __ __ __ _

WRITE(IflT62)(STRE(M),H1,10O)62 r,- MAT(3(-4t#5 2XoZWeE

HARK: I

CUT =CUT-19

___ ___ ___ ___ ___A-2 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

80 REAIDC O t4ARK,52) (NSN( I p=P)~evojp=,5,CACKp~p

OIVV.CALC(J)-STORE(J)Pt "ES=DIET'*DIFVV- - -HOLD (J ) HOLOCJ) +PL ACE

ft~auumuI -U -- -7-.

Tri4Al7K.IE irflIiwT- rn Tnj mfMARK21

PLACE=SORT(lfOLD(K )/CflUNTl)

63 FORMAT(tHO,4X#2OtfSTAfJDAR0 DEVIATION '2IOC4XpF6*'2).//)

IPOINTl

IF(HOLDCI).GE.PLACE) GOl TO A3

IPOINT=t

ALP=CflNSN1T( IPOINT)-~~~R4TEC-IUT641.. ALP, _____________________

64 FORMAT(IHO,44X,23H***** ALPHA SELECTED P F3.2p7H .. )

DO 84 Kz1,lPOINT.aK KOaiNTKP(LtNT+5 ____ __ ________

1tR.T-ZCIOUTP65) - ____ __ ________

65 FORMAT(lHI,49X#35aHSTN0LE EXPONVtITIAL SMOOTHING METHOO,/)

66 FORMAT(IHO,42XI4HFO.CAST DEI4ANC,31XPI3HACTUAL DEMAND#19XP

WRITEC IOUT,67)-4-RA T-( IH,*2X #2 1HAT IOlAL-STOCK( NUl8ERo,.4C,L0..@7nsO"t

I7HOCC s,7HJAN *7HFER o7HM4AR ),2XP1aHAVERAGE ERRORP//)

OIFF=O.O

A- 3

REA(AO 0 ARK,52) (NSN(I), 121,4 (R~s Vf(j),.JaZ, 15, (CALCCK),Kral,20,REAOC 1 tIARKP53)3( S"4OflTH(!Y. )v 1-t00JPO! NT=LPO TNTgo 84--N N21t5----LuN*10o$v:0rF*AOS C SMOOTH CJP&I N-T) -8-S VZ-CL-~jPO1NT=JP0INT4j

8A r aulr-1II r

1(OaSV0(J)pJ~llpl5)pOlrFF.O-TfR HA.TC 1H--, IXA'ID3H- 005j-~L4"--Z& lo X~ Z2)-XF,-

WCALCC 6)=OXFFSIX (1):LsTX )*(D~rr*nIEv)STXC2)=STX(2)+O1FF

V00

C ZEROING ARRAY CAIC IN PREPARATION FOR CALCULATED VALUES

Co UNT: I- P-14O&( I I=PLOAD +COUUJ.

10 CONTINUE--OUNTKOUIT---

CALCC i)-vi00 11 4:2,5

K -flSV(-8'OSO(-

VI2V/l1oO 'C 'AIC( -Q'i1

It CONTI*,4UE

r-SL THISP- W.A1.CULAIE.S. DEHANQ& 9.-TH TLZAsrSjUA ES METHODC

A-4 ---- I

P4sO.O

Un U+OBSVD(K)

PlsPlt(PEqI0O(K)*tJ3SVO(K~t))

P3xP3t-(PERIlDC(K)*oasvfl(x3))T~-4.*PER I Gfl(K *CBSVI) CK*4)-4

12 CONTINUE

CARDS NU!MREREfl t685o 1725p 1765# 1805,; 1845, FREVENT LEASTc SQU0ARES rRQX COINGa NECAITUE A NEGAT!'Tr IS 59*1 TO 0-

WICC(P+(CNNST*U))/99O.0)*C 1.0.CtNST)*12.O4Q

CALC(6)=WI11213-lRV~fl 1 4fl.ncvfl t I I

Quu/10 .0-~ f~~-W.-(CP 1.+ C CNNUS * U. 31-Z190.o1 to.L U0tN ST t.4Q-

IF(W2.LTooO) W2=0.0

JU-0FSV0(2)*0GSV0( 12)

W3:((P2+(CNIST'*J) )/990.O)*C1.0.CNNST),12.OQ

yCALC(a)=W3

QZU/10.0

IFCW4.LT.0*0) W4=I0.0C AL C-C-) 4 4U2UOR8SV(LI)+0BSVD( 14)

I5=C(P4*(CNNSr*U) )/99Oo.).C1.O.CNNST)*12.o+Q

CALC( 10)=W5

___________A- 5

85 CON~TINUE

69 FORIAT(IHI'55X,21HHOVIflq AVERAGE )ETHOD,/)

WRITEC IfUT#67)'

REA0(101!ARKE52)(NSNl(I),Izt,A4)p-(fRsvD(j)pj21,15),CCALC(K),K(1D2O)READCC1 4 A RKkS3 -)-C SRO GJ),J I*J -5~)-0)0 83 N=1,5

OIFF=DICFFABSCCAI.C(N)-fl9SVO(L))so COuTP-11 I

01 Vr=olrF/5 .0

* WRITEC inUT,68)CNSNC1),lIPi),(rAI.C(K),KtP5),

STV(2)=STV(2)+DIrFC AL' I CC17 ==FFINO:1

X31 .0-ALPYO-rALC(6)00 89 L=LPOINTPKPOIJT

CALC( IND)=Yl

[ND: INDO I

STVC2)=STVC2)/COllNT

WRITE(IOUTP 10)

DO 90 MARK:1..KGUMT

A- 6

00 91 N=6o-1

oIFF=OirF+ABSCCALC(N)-08SVD(L))

OlFrzorF/5oOWRT E-LO U T-a6 a C N SIN L* L: 1* 41 (CAL"kK,..K t.tLO~ 3

1(OJBSVO(.J),J=11lp19 ),rIFF

CALC(18)=DIFrr

DIFF=O.0

92 CONTINUE

BxB/6eO

C=C/ 12 .0

ALPH = ALPH *AL PH

C1:((3.*X)*C4...3.*ALPH4)/(2.*ALPi2))*C

~~Z:ALP*CALCCI ),(X*ZO)Wc.ALC-XltLl=Z

00 93 J=2P5

ZT=CALPli*CALCJ))+(X*Z)

Z=ZT

STW2)=STW(2)/COiINT-.OZ- Ff-AO .0---

____ ____ ___ ____ __ _ ____ ___ A- 7

WRITEC IOUIT#7t)AI izGRAT4!1Rl*9XA5WDCU9GLE EX09 NEtI1TI-AL SHOCTHI!IG H1h00 Q ' )

WRITEC lOtTP67)

REAQC1O HARK,52)(NSN(I),lIA),(OBSVOI(J),JztI5)P(CALCCK)DK21P20)An 95 ~-LzN410

95 CONTINUEC

CALC(t9)=D FFST y1t4.&T.YC1 ) CDIFV*DIFF)...-.STY2)=STY(2),O 1FF

WRITEC1O!MARK,52)(NSNCI),!IpA)(8SVO(.1),J=1,t5),(CALCCX)IKu1,2O) '

STY(2):STY(2)/COI!NT-. ~R ITtCZ OIJT-4)-- ___________________________

72 FGRMAT(1HI,4QX#35HTRIPt.E EXPONENTIAL SMCOTHING METHOD)

WRITEC IGUT,67)00 96 M'ARK-1-,-K-G2ITREAO(1O MARKP52)(NSN(I),lI,),(aBSsVD(J),J1,#15),(CALC(X),Ka1.20)

01 FF:0 FF. SC ALC (N )-nBSVD(N) )

OIF=IFF/5.O

ICOBSVD(J)PJzilP15)pOiFF -_____________________

STZ(l)=ST7(t)+(DlFF*flXFF)S 7(7':STi(21+oIFF _ _ _ __ _ _ _ __ _ _ __ _ _ _ __ _ _ _

96 CONTINUE

A-8

WRITECOUltTP73)

WRITEC TOUTP7'I)--4-FOA- i HOi'-2-1 H*KAT-I-ONAL--STOCK- NllR-P-3#74

17HA JANp5Xp7HA rr8psx, 7HA ?4AR,5XP*3W-NOV,4X,3HDECCAX,3HJA~i,4X*

00 98 MARK=1,KOUt:T

REA0C11 NARKS3)(S)40OTti(J)pJj,50 )

V1=V/10.0

A=ALPH

00L.E.4 100 to 03

TEST=41..2-STDV

V2:Vl-TEST2

zr CBSVO(L.,) .E.V) A160 VO12.G.2) =.

XT(A'OOSVOC) )+(aTO)10T6R2-1,*S-!E.:

1-9 -.I-

STORE C2*K)=XT

V=CV-oBSVoCK) ) .OVSVO(L.1)

rFCSTORE(2*KmI)*FQ.1.O) ,,:.A

IFCSTORE(2*K-1)*Eg..5) A=ALPH

C IF YOU WANT ALPHA Tfl SKIP .8 VALUE,- CHANGE THE .8 VALUE CMC c-UG EISTC ta~~CC~ NU~ER 1555 - _ _

!XCLmLE Ii) QG TO 5O

DIFF:O.O

DO 104 Mi2,1O,2

DICF=OirF/5.0

7-5 aZA-T2X43H.O3A41OCF6-),4K 5E . 91.5XAE& A)STZA(I )=STZA( I )(DIFF*nIFF)

SMOOTH( 1):O 1FF

WSTV( TV) =SV(3)(CALC( 17).'ST'(2))*(CALCC 17)-STV(2))) -

STX(3):STX(3)sC(CALC(I6)inSTX(2))*(CALC(16)-STX(2)))

STZC3)=STZ( 3)+C((ALC(2O))-ST7(2))*(CALC(2O)-STZ(2)))

STZA(2):=,TZA(2)/COUNT00 1-05 llAQK1fiz ,KONTREADC1i !ARK53)(SHIOflT(J),.J=1,'O)

105 CONTINUC-COUIC4CUT ItsKOUN.-T 9 1- - ______________________

A-10

WRITEC IOtiT,76)

lERROR,IOXp1CHMEAfl rORCAST ERRORpi2XpflHVARIANCEp///)SMCy3I12- StCll /C QP1NT-L---STVC(1)cSTV(1 )/Cflh'NT

77 FORMAT(1HO,i0XI4HMOIlttG AVERAI,18X,3(F,3.2,2OX)'-///)

STwCI )=STW(lI)/C0t'NTAR TtE Ci1U T-A 8) C S.T WCJLdz.l)-

7?i t0RMAT(1HO,1OXpl3fiLEAST SQUARES,19X,.3(F8-2p20X),//

STX( I)=STX( I)/C.0VNT

79 FORMAT( HOD5X,28HSINALE EXPONENTIAL SHOOTHINGp9X*3(F8*2v2X)/g#STY C 3 -S-TY 341/CGUNT-1-~-STY( i)=STY( t )/ClUNT

30 FORHAT( iHO,5X,28HOUUflLE EXPONENTIAL SMOOTHINC,9Xi13CP8e2#2OK)*///)

STZ(l)=STZC I /Cal1NT

31 FORMAT( lHOSXv2EitiTRIPLE EXPONENTIAL SMOOTHING v9XP3(Fa.2p2OX)#///)ST AC )- --%.Z ( 1 C I1TSTZA(I z STZA(I)/COUINT

32 FORMATCIHoplX*37H1ADAPTIVE SINGLE. EXPONENTIAL SMOOTHING v 4Xv

WRITE(IOIJT,33)

STOP

0 :14 P.M. ASR 5.4 USItIA 73/345 XVORTN COMPILERA.T -32-P-&I, 1 Ft AG- FR~

0 DATA = 6t?4 TEM4PORARIES 7 2 CODE 27900 DIGITS

_____ _ _______A-li

I

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