business forecasting chapter 4 odd solutions

8/9/2019 Business Forecasting Chapter 4 Odd Solutions

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CHAPTER 4

MOVING AVERAGES AND SMOOTHING METHODS

ANSWERS TO ODD-NUMBERED PROBLEMS AND CASES

1. Exponential smoothing

3. Moving average

5. Winters’ three-parameter smoothing procedure

7.

Price AVER1 FI!1 RE!I11"#$" % % %

1"' % % %1() 1&*)) % %1+#&" 1$*)) 1&*)) -)#"')))1,$ 1+$$ 1$*)) )#)&)))1"#"& 1+''+ 1+$$ 1#&)''+1"#*1 1"#$)'+ 1+''+ )#+,$$$()#'$ ()#),)) 1"#$)'+ 1#$($$$1"#+& 1"#"+$$ ()#),)) -)#(')))(1#(* ()#**$$ 1"#"+$$ 1#(+''+(1#1& ()#+$'+ ()#**$$ )#'(''+((#1, (1#*($$ ()#+$'+ 1#,)$$$

Accurac MeasuresMAPE. ,#'$1" MA/. )#",(( M!E. 1#1+(&

he na0ve approach is etter#

9. a# 2 c3 d3 e3 4 $-month moving-average 5!ee plot elo6#7

Month 8ield MA Forecast Error1 "#(" % % %

( "#"" % % % $ 1)#1' "#&1$ % %

, 1)#(* 1)#1$$ "#&1$ )#,$+ * 1)#'1 1)#$,) 1)#1$$ )#,++ ' 11#)+ 1)#',$ 1)#$,) )#+$) + 11#*( 11#)'+ 1)#',$ )#&++ & 11#)" 11#((+ 11#)'+ )#)($ " 1)#&) 11#1$+ 11#((+ -)#,(+ 1) 1)#*) 1)#+"+ 11#1$+ -)#'$+ 11 1)#&' 1)#+() 1)#+"+ )#)'$

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1( "#"+ 1)#,,$ 1)#+() -)#+*)

Accurac MeasuresMAPE. ,#*&+* MA/. )#,"11 M!E. )#$1"$ MPE. #'"),

Forecast 4or month 1$ 59an#7 is 1)#,,$

# 2 c3 d3 e3 4 *-month moving-average 5!ee plot elo6#7

Month 8ield MA Forecast Error 1 "#(" % % %

( "#"" % % % $ 1)#1' % % %

, 1)#(* % % % * 1)#'1 1)#)') % %

' 11#)+ 1)#,1' 1)#)') 1#)1) + 11#*( 1)#+(( 1)#,1' 1#1),

& 11#)" 1)#")& 1)#+(( )#$'& " 1)#&) 11#)1& 1)#")& -)#1)&

1) 1)#*) 1)#""' 11#)1& -)#*1& 11 1)#&' 1)#"*, 1)#""' -)#1$'

1( "#"+ 1)#',, 1)#"*, -)#"&,

Accurac MeasuresMAPE. *#*&$) MA/. )#'),) M!E. )#*()( MPE. #+1))

Forecast 4or month 1$ 59an#7 is 1)#',,

(


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g# :se $-month moving average 4orecast. 1)#,,$$

11. !ee plot elo6#

Month /emand !mooth Forecast Error

1 ()* ()*#))) ()*#))) )#)))) ( (*1 (())) ()*#))) ,'#)))) $ $), (''#))) (())) +'#)))) , (&, (+*#))) (''#))) 1))))

* $*( $1$#*)) (+*#))) ++#))))

' $)) $)'#+*) $1$#*)) -1$#*))) + (,1 (+$#&+* $)'#+*) -'*#+*))

& (&, (+"$& (+$#&+* 1)#1(*) " $1( ("*#,'" (+"$& $$#)'(*

1) (&" ("(#($, ("*#,'" -'#,'&& 11 $&* $$'1+ ("(#($, "(#+'*' 1( (*' ("+#$)" $$'1+ -&(#'1+(

Accurac MeasuresMAPE. 1,#'+ MA/. ,$#,, M!E. (",$#(,

Forecast 4or month 1$ 59an# ())+7 is ("+#$)"

$


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13. a# α ; #,

Accurac MeasuresMAPE. 1,#)* MA/. (,#)( M!E. 11+,#*)

Forecast 4or


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15. A time series plot o4 @uarterl Revenues and the autocorrelation 4unction sho6that the data are seasonal 6ith a trend# A4ter some experimentation3 Winters’multiplicative smoothing 6ith smoothing constants ? 5level7 ; )#&3 5trend7 ; )#1and B 5seasonal7 ; )#1 is used to 4orecast 4uture Revenues# !ee plot elo6#

Accurac Measures

MAPE $#&MA/ '"#1M!E 111,'#,

Forecasts


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An examination o4 the autocorrelation coe44icients 4or the residuals 4romWinters’ multiplicative smoothing sho6n elo6 indicates that none o4 themare signi4icantl di44erent 4rom Cero#

17. a# he 4our-6ee> moving average seems to represent the data a little etter#

Dompare the error measures 4or the 4our-6ee> moving average in the 4igure elo66ith the 4ive-6ee> moving average results in Figure ,-,#

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# !imple exponential smoothing 6ith a smoothing constant o4 ? ; #+ does a etter o o4 smoothing the data than a 4our-6ee> moving average as udged the uni4orml smaller error measures sho6n in the plot elo6#

19. a# he results o4 olt’s smoothing 6ith ? 5level7 ; #" and 5trend7 ; #1 4or!outh6est Airline’s @uarterl income are sho6n elo6# A plot o4 the residualautocorrelation 4unction 4ollo6s# It appears as i4 olt’s procedure represents thedata 6ell ut the residual autocorrelations have signi4icant spi>es at the seasonal

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lags o4 , and & suggesting a seasonal component is not captured olt’smethod#

# Winters’ multiplicative smoothing 6ith ? ; ; B ;#( 6as applied to the @uarterlincome data and the results are sho6n in the plot elo6# he 4orecasts 4or the4our @uarters o4 ())) are.


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*1 1&1#,',*( 11+#"&*

he 4orecasts seem reasonale ut the residual autocorrelation 4unction elo6 hasa signi4icant spi>e at lag 1# !o although Winters’ procedure captures the trend andseasonalit3 there is still some association in consecutive oservations notaccounted 4or Winters’ method#

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CASE 4-1: THE SOLAR ALTERNATIVE COMPANY

his case provides the student 6ith an opportunit to deal 6ith a 4re@uent real 6orld prolem. small data sets# A plot o4 the t6o ears o4 data sho6s oth an up6ard trend and seasonal pattern# he 4orecasting model that is selected must do an accurate o 4or at least three months intothe 4uture#

Averaging methods are not appropriate 4or this data set ecause the do not 6or> 6hen datahas a trend3 seasonalit3 or some other sstematic pattern# Moving average models tend to smooth outhe seasonal pattern o4 the data instead o4 ma>ing use o4 it to 4orecast#

A naive model that ta>es into account oth the trend and the seasonalit o4 the data might6or># !ince the seasonal pattern appears to e strong3 a good 4orecast might ta>e the same value itdid in the corresponding month one ear ago or 8tG1 ; 8t-11#

o6ever3 as it stands3 this 4orecast ignores the trend# Hne approach to estimate trend is to calculatethe increase 4rom each month in ())* to the same month in ())'# As an example3 the increase 4rom9anuar3 ())* to 9anuar3 ())' is e@ual to 581$

- 817 ; 51+ - *7 ; 1(#

A4ter the increases 4or all 1( months are calculated3 the can e summed and then divided 1(# he 4orecast 4or each month o4 ())+ could then e calculated as the value 4or the same month in())' plus the average increase 4or each o4 the 1( months 4rom ())* to the same month in ())'#Donse@uentl3 the 4orecast 4or 9anuar3 ())+ is

8(* ; 1+ G 51+ - *7 G 51, - '7 G 5() - 1)7 G 5($ - 1$7 G 5$) - 1&7 G 5$& - 1*7 G 5,, - ($7 G 5,1 - ('7 G 5$$ - (17 G 5($ - 1*7 G 5(' - 1(7 G 51+ - 1,7JK1(

8(* ; 1+ G1(

1,& ; 1+ G 1( ; ("

he 4orecasts 4or ())+ are. 9an (" Fe (' Mar $( Apr $* Ma ,( 9un *) 9ul *' Aug *$ !ep ,* Hct $* Lov $& /ec ("

Winters’ multiplicative method 6ith smoothing constants ? ; #13 ; #13 B ; #$ seems torepresent the data 4airl 6ell 5see plot elo67 and produces the 4orecasts.

Month Forecast9anK())+ 1"#&FeK())+ 1)MarK())+ ('#&

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AprK())+ $(#)MaK())+ ,(#,9unK())+ ,*#&9ulK())+ *,AugK())+ *"!epK())+ ,+#'

HctK())+ $$#+ LovK())+ $$#*/ecK())+ ()

he na0ve 4orecasts are not unreasonale ut the Winters’ 4orecasts seem to have captured theseasonal pattern a little etter3 particularl 4or the 4irst $ months o4 the ear# Lotice that i4 the trendand seasonal pattern are strong3 Winters’ smoothing procedure can 6or> 6ell even 6ith onl t6oears o4 monthl data#

CASE 4-2: MR TUX

his case sho6s ho6 several exponential smoothing methods can e applied to the Mr# uxdata# 9ohn Mos tries simple exponential smoothing and exponential smoothing 6ith adustments4or trend and seasonal 4actors3 along 6ith a three-month moving average#

!tudents can egin to see that several 4orecasting methods are tpicall tried 6hen animportant variale must e 4orecast# !ome method o4 comparing them must e used3 such as thethree accurac methods discussed in this case# !tudents should e as>ed their opinions o4 9ohns progress in his 4orecasting e44orts given these accurac values# It should e apparent to most thatthe degree o4 accurac achieved is not su44icient and that 4urther stud is needed# !tudents should e reminded that the are loo>ing at actual data3 and that the prolems 4aced 9ohn Mos realloccurred#

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1# H4 the methods attempted3 Winters’ multiplicative smoothing 6as the est method 9ohn4ound# Each 4orecast 6as tpicall o44 aout (*3&(*# he error in each 4orecast 6as

aout ((N o4 the value o4 the variale eing 4orecast#

$# 9ohn should examine plots o4 the residuals and the residual autocorrelations# I4 Winters’ procedure is ade@uate3 the residuals should appear to e random# In addition3 9ohn can

examine the 4orecasts 4or the next 1( months to see i4 the appear to e reasonale#

CASE 4-3: CONSUMER CREDIT COUNSELING

1# !tudents should realiCe immediatel that simpl using the asic naive approach o4using last period to predict this period 6ill not allo6 4or 4orecasts 4or the rest o41""$# !ince the autocorrelation coe44icients presented in Dase $-$ indicatesome seasonalit3 a naive model using April 1""( to predict April 1""$3 Ma 1""( to predict Ma 1""$ and so 4orth might e tried# his approach produces the error

measures

MA/ ; ($#$" M!E ; &'1#$, MAPE ; 1"*

over the data region3 and are not particularl attractive given the magnitudes o4 the ne6client numers#

$ !ince the data have a seasonal component3 Winters’ multiplicative smoothing procedure 6ith smoothing constants ? ; ; B ;#( 6as tried# For these choices.MA/ ; 1"#("3 M!E ; *,*#,1 and MAPE ; 1'#+,# For smoothing constants ? ; #*3 ; B ; #13 MA/ ; 1'#",3 M!E ; ,*1#(' and MAPE ; 1,#$)#

*# :sing Winters’ procedure in ,3 the 4orecasts 4or the remainder o4 1""$ are.

Month ForecastAprK1""$ 1,&MaK1""$ 1,19unK1""$ 1,&9ulK1""$ 1,1AugK1""$ 1,$!epK1""$ 1$'HctK1""$ 1*" LovK1""$ 1,'/ecK1""$ 1('

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CASE 4-4: MURPHY BROTHERS FURNITURE

1# Lo ade@uate smoothing model 6as 4oundO A Winters’ multiplicative model using? ; #$3 ; #( and B ; #1 6as deemed the est ut there 6as still some signi4icantresidual autocorrelation#

$# ased on the 4orecasting methods tested3 actual Murph rother’s sales data should eused# A plot o4 the results 4or the est Winters’ procedure 4ollo6s#

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An examination o4 the autocorrelation coe44icients 4or the residuals 4rom this Winters’model sho6n elo6 indicates that none o4 them are signi4icantl di44erent 4rom Cero#

o6ever3 9ulie decided to use the na0ve model ecause it 6as ver simple and she couldexplain it to her 4ather#

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

his case is designed to emphasiCe the use o4 suective proailit estimates in a4orecasting situation# he methodolog used to generate revenue 4orecasts is oth appropriateand accuratel emploed# he >e to ans6ering the @uestion concerning the accurac o4 the proections hinges on the accurac o4 the assumptions made and estimates used# Examinationo4 the report indicates that the analsts 6ere conservative each time the made an assumption or

computed an estimate# his is proal one o4 the maor reasons 6h the Pro4essionalMar>eting Associates’ 5PMA7 4orecast is consideral lo6er# !ince 6e do not >no6 ho6 theaccountant proected the numer o4 procedures3 it is di44icult to determine 6h his revenue proections6ere higher# o6ever3 it is reasonale to assume that his 4orecast o4 the numero4 cases 4or each tpe o4 procedure 6as not nearl as sophisticated or thorough as PMAs#here4ore3 the recommendation to management should indicate that the PMA 4orecast3 6hile proal on the conservative side3 is more li>el to e accurate#

/o6nto6n Radiolog evidentl agreed 6ith PMAs 4orecast# he decided not to purchase a "3&)) series D scanner# he also decided to purchase a less expensive MRI#Finall3 the decided to otain outside 4unding and did not resort to an tpe o4 pulic o44ering#he uilt their ne6 imaging center3 purchased an MRI and have created a ver success4ul

imaging center#

CASE 4-: WEB RETAILER

1# he time series plot 4or Hrders sho6s a slight up6ard trend and a seasonal pattern6ith pea>s in /ecemer# ecause o4 the relativel small data set3 the autocorrelationsare onl computed 4or a limited numer o4 lags3 ' in this case# Donse@uentl 6ithmonthl data3 the seasonalit does not sho6 up in the autocorrelation 4unction# here

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is signi4icant positive autocorrelation at lag 13 so Hrders in consecutive months arecorrelated#

he time series plot 4or DPH sho6s a do6n6ard trend ut a seasonal component isnot readil apparent# here is signi4icant positive autocorrelation at lag 1 and theautocorrelations die out relativel slo6l# he DPH series is nonstationar and

oservations in consecutive time periods are correlated#

$# !imple exponential smoothing 6ith ? ; #++ 5the optimal ? in Minita7 represents thethe DPH data 6ell ut3 li>e an Qaveraging procedure3 produces 4lat-line 4orecasts#Forecasts o4 DPH 4or the next , months are.

Month Forecast =o6er :pper 9ulK())$ )#1),* )#)+&+ )#1$)$

AugK())$ )#1),* )#)+&+ )#1$)$!epK())$ )#1),* )#)+&+ )#1$)$HctK())$ )#1),* )#)+&+ )#1$)$

he results 4or simple exponential smoothing are pictured elo6# here are no

signi4icant residual autocorrelations 5see plot elo67#

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*# It seems reasonale to 4orecast Dontacts directl i4 the data are availale#Multipling a 4orecast o4 Hrders a 4orecast o4 DPH to get a 4orecast o4 Dontactshas the potential 4or introducing additional error 5uncertaint7 into the process#

CASE 4-!: SOUTHWEST MEDICAL CENTER

1# Autocorrelation 4unction 4or total visits suggests time series is nonstationar5since autocorrelations slo6 to die out7 and seasonal 5relativel large autocorrelationat lag 1(7#

$# I4 another 4orecasting method can ade@uatel account 4or the autocorrelationin the otal Visits data3 it is li>el to produce Qetter 4orecasts# his issueis explored in suse@uent cases#

CASE 4-": SURTIDO COO#IES

1# 9ame learned that !urtido Doo>ie sales have a strong seasonal pattern5sales are relativel high during the last t6o months o4 the ear3 lo6 duringthe spring7 6ith ver little3 i4 an3 trend 5see Dase $-*7#

$# Winters’ multiplicative smoothing 6ith ? ; ; B ; #( seems to represent thedata 4airl 6ell and produce reasonale 4orecasts 5see plot elo67# o6ever3there is still some signi4icant residual autocorrelation at lo6 lags#

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Month Forecast =o6er :pper 9unK())$ '*$(*, "1$*1 1(1*1*+ 9ulK())$ +1(1*" 1,1,*$ 1(&(&'* AugK())$ '**&&" +*$'& 1($',11 !epK())$ 1*$(",' ",1',+ (1(,(,*

HctK())$ 1+1)*() 11)+*$$ ($1$*)+ LovK())$ (1$$&&& 1*1&$*, (+,",(1 /ecK())$ 1")$*&" 1(+,+)( (*$(,+'

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business forecasting chapter 4 odd solutions

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