or final project back work
TRANSCRIPT
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THE MCS CORPORATION PROJECTAMEYA SAMPAT & HUY NGUYEN
Operation Research Methods
Professor Sharkey
1!"#!"1
Introduction/Problem BackgroundThe M$S $orporation% prodcers of the toy '(ino)a**%+ is interested in ,a-i,i.in/ pro0ts
hi*e ,ini,i.in/ prodction cost2 Actin/ as ad3isors to the co,pany% or /rop as char/ed to
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0nd the ansers to three 4estions to he*pM$S reach the ,ost cost5e6cient prodction ,ethod2
The 0rst 4estion as 'Sho*d they re,o3e the )a*anced po*icy that re4ires each of M$S
prodction faci*ities to prodce the sa,e n,)er of toys7+ The second 4estion as% 'Sho*d
M$S $orporation insta** 8astProd% a syste, that o*d a**o for faster and cheaper prodction of
(ino)a**s7+ The third 4estion in3o*3es a /reen initiati3e that the M$S $orporation is prsin/2
9n order to *oer the car)on e,ission to )e*o the per,itted *i,it of 1" ,etric ton% the co,pany
has an option to insta** a certain n,)er of so*ar pane*s per faci*ities2 The insta**ation o*d he*p
redce a cost that certain faci*ities o*d incr that o*d dri3e o3era** cost2
To anser the 4estions% an ana*ysis as done in e-ce* sin/ Operations Research
Methods hich co*d )e app*ied to de3e*op a decision choice2 The fo**oin/ report detai*s the
0ndin/s of the report and i** a*k thro/h the steps sed to co,e to that decision2
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Remoal o! t"e Polic#$e%ning t"e Problem
$rrent*y% M$S operates nder the po*icy that each faci*ity i** prodce the sa,e n,)er of
(ino:a** toys each ,onth2 9n other ords% the faci*ities are +)a*anced+ in that each i** prodces
e-act*y 1!; of M$S
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Operation ith :a*ance Po*icy
Or /rop sed E-ce* So*3er to ca*c*ate ,onth*y prodction cost and shippin/ cost2 :e*o is an
e-a,p*e of the cost ca*c*ation >for (ece,)er? and an e-p*anation for ho e for,*ated the
cost2
Troy% NY'nit&/"
our
Energ#()*+/"
our
Energ#
Co&t/Hr
$eci&ion
,ariable&
(Hr&+
Time-imit(Hr&+ Co&t
No. o!Product
&
Prodine 1 # "" "21B
DB21DB1DB 1#"
BBB2BBBBB#
B112111111
Prodine BB
;BD2;;;;;;
1C"2;C";C
Prod
ine ; 1 CB
;B1;2DD
DDDF
DB21DB1D
B
(e,a
nd
Tota* No2 of
Prodcts@C##2####
#CC##2#
##CNeark%NProdine 1 B 1" "21D
FB2D;;;;; 1#" ;#2B
=CF21####C
Prodine BB
=;FD2CB1F12####
#CProdine ; 1 CB =C=;2CB
FB2D;;;;;;
(e,and
Tota* No2 of
Prodcts@
C##2####
#C
C##2#
##CHarris)r/% PA
Prodine 1 = " "2"F
1"2 1#" "=
="D2DDDDDF
Prodine ; ;B
1#;"#2####
#CProdine ; "2B ;"" C#"
B121111111
(e,and
Tota* No2 of
Prodcts@C##2####
#CC##2#
##C
Tota*cost
0123.4443
Tota* Spp*y ;""
Tota* (e,and ;""
Table 5b6 Sam7le o! Production Co&t Calculation !or Balance Polic# O7eration
Para,eter e-p*anation@
(ecision 3aria)*es@ the ti,e each prodction *ine in a partic*ar factory operates2 They are
deter,ined )y E-ce* So*3er as an opti,a* so*tion to ,ini,i.e prodction cost2 Since M$S
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ses the e4a* 'ear and tear+ po*icy% a** three prodction *ines in each factory i** ha3e
the sa,e operatin/ ti,e2 Each of the decision 3aria)*es has to )e s,a**er than the 1#"
hor ti,e *i,it% the ,a-i,, a,ont of ti,e a ,achine can operate e3ery ,onth2
$ostEnergy
Hour
EnergyCost
Hour DecisionVariable.
$ost is the prodction cost for each
prodction *ine2 Tota* cost is the s, of a** costs fro, a** prodction *ines2
No2 of Prodcts Unit aria)*es I (ecision aria)*e2 The tota* n,)er prodcts of each
factory has to )e e4a* to e-act*y 1!; of the ,onth*y de,and de to )a*ance po*icy Tota* cost is hat E-ce* So*3er tries to ,ini,i.e2 9t is the prodction cost for that ,onth >in
this case (ece,)er?2
Para,eter e-p*anation@
J Units shipped@ the a,ont of prodcts shipped fro, a partic*ar factory to a partic*ar
destination2 These are deter,ined )y E-ce* So*3er to ,ini,i.e the tota* shippin/ cost2 The
tota* a,ont of nits shipped fro, each factory has to )e e4a* to the spp*y of that
factory >i2e2 Tota* No2 of prodcts in Ta)*e 1)?2 The tota* a,ont of nits shipped to each
retai*er has to )e e4a* to the de,and of that retai*er2 J of Trips@ the n,)er of trips it takes to ship the a,ont of prodcts fro, one partic*ar
factory to partic*ar retai*er2 E-a,p*e@ it takes trips to ship ;"" prodcts fro, Harris)r/
to Pitts)r/h2 Each trip can on*y carry a ,a-i,, of B" prodcts2 These are a*so
deter,ine )y E-ce* So*3er2
Tota* $apacity J of Trips I B"2 The J of Units shipped a*ays has to )e s,a**er than
this n,)er2
Tota* Unit Shippin/ cost S, >J Units shipped I Shippin/ $osts Per Unit?2 The shippin/
cost per nit is pro3ided )y M$S in proKect data2
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Tota* Trip cost S, J of Trips I Shippin/ $osts Per Trip?2 The shippin/ cost per trips
pro3ided )y M$S in proKect data2
Tota* $ost Tota* Unit Shippin/ cost L Tota* Trip cost2 This is the shippin/ cost and the
o)Kecti3e fnction that E-ce* So*3er tries to ,ini,i.e2
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8or each ,onth% ta)*es *ike a)o3e are created to ca*c*ate the operation cost2 The operation cost
is the s, of shippin/ cost and prodction cost2 The res*ts are recorded in the Ta)*e 1d )e*o2
Tota*(e,and
Prodction$ost> ith:a*ance?
Shippin/cost>ith:a*ance?
Operationcost ith:a*anceMonth
anary 1=B 1CFF12= 11D2B# 1F"F2FD
8e)rary 1=CB1D#2#F 1;"B21# 1FFC2DB
March 1BB 1FB;2FC 1;F2# "BD;2;
Apri* 1BCB 1FDDB2B 1;D=2F# 1C"21
May 1=B" 1D;"C2"# 1;==2" 1F#B12B
ne 1=D" 1D#DB2D 1;=C21 "";2F=
*y 1=;" 1D"B=2B= 1;B#2D; 1F=112;C
A/st 1B"" 1DF;D2;; 1;;2=D "C"2D1
Septe,)er 1;"" 1#=1;2 1CC2;C 1C#F"2BF
Octo)er 1=CB 1D#2#F 1;""2CD 1FF;2=D
No3e,)er 1DCB ;#C2F 1B"D2D" B1D12C
(ece,)er ;"" F";D2CD 1DB;2"; ;"DF12D1
Tota*
;C=D#2C" 1#BBD2BB B="=B2B
Table 5d6 Mont"l# O7eration co&t 8it" Balance Polic#
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Operation ithot :a*ance Po*icy
Usin/ the si,i*ar techni4e as descri)ed a)o3e ith operation ith )a*ance po*icy% ith the
e-ception of the de,and constraint >the tota* n,)er prodcts of each factory has to )e e4a* to
e-act*y 1!; of the ,onth*y de,and de to )a*ance po*icy5Ta)*e 1) para,eter e-p*anation?% e
o)tained the operation cost ithot the )a*ance po*icy2 9n addition% the prodction cost and
shippin/ are done si,*taneos*y2 The e-a,p*e of an inte/rated ,ode* >for *y? is shon )e*o2
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The )a*ance po*icy is re,o3ed so each factory is free to prodce as ,any prodcts as they can
ithot de,and constraints as *on/ as their tota* prodction e4a*s the tota* de,and2 The
spp*y constraints >the tota* a,ont of nits shipped fro, each factory has to )e e4a* to the
spp*y of that factory% i2e2 tota* no2 of prodcts? sti** re,ain2 8or e-a,p*e% Troy prodces ;"
prodcts% hich ,eans on*y a ,a-i,, of ;" nits can )e shipped fro, Troy2 Since the ,ode*
is no inte/rated% the o)Kecti3e fnction is no to ,ini,i.e the Tota* $ost > Unit shippin/ cost
L Tota* Trip $ost L Tota* Prodction $ost? instead of to separate o)Kecti3e fnction as in )a*ance
po*icy operation2 This Tota* $ost is ca*c*ated for each ,onth and recorded in the ta)*e )e*o2
Tota* (e,and
Operation cost ithot
:a*ance Po*icyMonth
anary 1=B 1B;BC2FB
8e)rary 1=CB 1BDD2F"
March 1BB 1#=C;21B
Apri* 1BCB 1C=D21;
May 1=B" 1B#C#2;F
ne 1=D" 1BFD121B
*y 1=;" 1B="#2
A/st 1B"" 1#1F2"D
Septe,)er 1;"" 1;CD12B
Octo)er 1=CB 1BF;F2CC
No3e,)er 1DCB 1""F2=1
(ece,)er ;"" #;12D;
Tota* 1DD1" "B#12
Table 5!6 Mont"l# O7eration co&t 8it"out
Balance Polic#
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Com7ari&on
8ro, Ta)*e 1d and 1f% the ta)*e )e*o is created in order to co,pare the cost re/ardin/ )a*ance
po*icy2
Tota*(e,a
nd
Operationcost ith:a*ancePo*i3icy
Operation costithot :a*ance
Po*icy
Percentof
sa3in/sMonth
anary 1=B 1F"F2FC#C 1B;BC2FB;; "21
8e)rary 1=CB 1FFC2DB 1BDD2F "2;
March 1BB "BD;2;== 1#=C;21B;; "2"
Apri* 1BCB 1C"21 1C=D21; 1D2F
May 1=B" 1F#B12B11 1B#C#2;D#C "2
ne 1=D" "";2F== 1BFD121=C; "2
*y 1=;" 1F=112;CF 1B="#21# "2#
A/st 1B"" "C"2D1;; 1#1F2"D "21
Septe,)er 1;"" 1C#F"2BF11 1;CD12B;; 21
Octo)er 1=CB 1FF;2=C#C 1BF;F2C##C "2"
No3e,)er 1DCB B1D12C1#C 1""F2="#C 1#2#
(ece,)er ;"" ;"DF12D"DF #;12D#C 1=2D
Tota* B="=B2=# "B#12 1F2
Tota* Sa3in/s 93439.1:3 50.;Table 5g Total O7eration Co&t& Com7ari&on regarding
Balance Polic#
Conclu&ion
After re3iein/ ta)*e 1/% it as deter,ined that the ,ost cost eecti3e choice o*d )e to
operate the faci*ities ithot )a*ance2 here )a*ance is de0ned as 'prodction *e3e*s are thesa,e2+ The 0ndin/s indicate that the tota* cost of operatin/ ith )a*ance is B=%"=B2= and the
tota* ithot )a*ance )ein/ "B%#122 This o*d ,ean a sa3in/s of =D%CD=2"2 Ths% a
choice to choosin/ a prodction syste, ithot )a*ance po*icy o*d )e the )est ay of
reachin/ M$S $orporation
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In&tallation o! $e%ning T"e Problem
As ,entioned )efore% M$S $orporation as cha**en/ed ith the 4estion of hether or not
to insta** a ne syste, ca**ed '8astProd2+ This syste, as desi/ned not on*y to rep*ace the
e-istin/ prodction *ine% )t it as desi/ned to trn ot (ino:a**s at a faster rate and conser3e
,ore ener/y2 A*tho/h the choice to insta** this ne syste, see,s tri3ia*% the decision asco,p*icated )y the re4ire,ent that M$S $orporation had to prchase a ne syste, for each of
their factories in order for any dea* to occr2
9n order to ,ake any decisions and ha3e ,ore *e3era/e drin/ a ne/otiation% M$S
$orporation re4ested that their 'reser3e price+ )e ca*c*ated2 The 'reser3e price+ as the
,a-i,, a,ont that M$S $orporation as i**in/ to pay for a certain type of contract2 :e*o
are the series of steps sed to ca*c*ate and the reser3e prices of the to types of syste,s2
Anal#&i&
e took si,i*ar approach as e did ith the re,o3a* of po*icy pro)*e,2 The operation costs ereca*c*ated sin/ E-ce* So*3er2 A** the prodction characteristics ere chan/ed to 8astProd
characteristics2 An e-a,p*e of this is shon in the ta)*e )e*o2
Troy% NY'nit&/"
our
Energ#()*+/"
our
Energ#
Co&t/Hr
$eci&ion
,ariable&
(Hr&+
Time-imit(Hr&+ Co&t
No. o!Produc
t&
Prod ine1 D ;"" "21B
;12F===== 1#"
1;=12#####C
BB2BBBBB#
Prod ine
D ;""
1;=12###
##C
BB2BBB
BB#Prod ine; D ;""
1;=12#####C
BB2BBBBB#
(e,and
Tota* No2 of
Prodcts@C##2###
##CC##2#
##CNeark%NProd ine1 D ;"" "21D
;12F===== 1#"
1;=12#####C
BB2BBBBB#
Prod ine D ;""
1;=12#####C
BB2BBBBB#
Prod ine; D ;""
1;=12#####C
BB2BBBBB#
(e,and
Tota* No2 of
Prodcts@C##2###
##CC##2#
##CHarris)r/% PAProd ine1 D ;"" "2"F
;12F===== 1#"
1;=12#####C
BB2BBBBB#
Prod ine D ;""
1;=12#####C
BB2BBBBB#
Prod ine; D ;""
1;=12#####C
BB2BBBBB#
(e,and
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Tota* No2 of
Prodcts@C##2###
##CC##2#
##C
Tota*cost 514:
Tota* Spp*y ;""
Tota* (e,and ;""Table a6 Sam7le o! =a&t Production Co&t Calculation !or Balance Polic#
O7eration !or $ecember
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To /et an nderstandin/ of ho ,ch M$S $orporation as i**in/ to pay% the cost of operatin/
the pre3ios syste, for three years ith )a*ance and ithot )a*ance as ca*c*ated and
co,pared to the cost of operatin/ the 8astProd ith )a*ance and ithot )a*ance2 The ta)*e
)e*o shos or 0ndin/s2
NormalO7eration
co&t 8it"BalancePolic#
=a&tProdO7eration
co&t 8it"BalancePolic#
NormalO7eration
co&t 8it"outBalancePolic#
=a&tProd
O7erationco&t 8it"
Balance Polic#Month
anary 1F"F2FC##C D#FF2D1 1B;BC2FB;;; #CD2D"D;;;
8e)rary 1FFC2DB F"=D2F"BBB# 1BDD2F #FC12=B
March "BD;2;=== F;;B2B1 1#=C;21B;;; C1=2D1##C
Apri* 1C"21 F#B;2C1 1C=D21; C;#B2F1D;;;
May 1F#B12B111 DFB#2#FBBB# 1B#C#2;D##C #DD12B;###C
ne "";2F== F11C211CCD 1BFD121=C;; #FD=2=F
*y 1F=112;CDF DD#=2;D=== 1B="#21# #C1;21"D##C
A/st "C"2D1;;; F"C2=D 1#1F2"D C";12=###C
Septe,)er 1C#F"2BF111 D1"2;#DDDF 1;CD12B;;; #1B#2;;###C
Octo)er 1FF;2=C##C F"==2B; 1BF;F2C###C #FBF2FBD;;;
No3e,)er B1D12C1##C 11;B2BB 1""F2="##C DC"F2#F1##C
(ece,)er ;"DF12D"DDF 1;FD2";111 #;12D##C 1"B;B2F=
Tota* B="=B2=BD 11B;112"=BD "B#12 DD1D121"=
;5year Tota* 4?52:.4242 29:022.5242 ?5:432.?? ?9:92.25; Year(ierence =1#"2# ;B1="2;=D
Ta)*e Operation $ost $o,parison Ta)*e
9n order to deter,ine the reser3e price% the prodction costs >for nor,a* operation ith
)a*ancin/% 8astProd ith )a*ancin/% nor,a* operation ithot )a*ancin/ and 8astProd ithot
)a*ancin/? ere ca*c*ated )y s,,in/ the ,onth*y rate for a 1 ,onth period% ith n,)ers at
the top of the co*,n representin/ anary and the )otto, )ein/ (ece,)er2 This then as
,*tip*ied )y three as the co,pany re4ested a three year period to )e ana*y.ed2 Then the
dierence as taken of the respecti3e syste,s and their ne conterparts to deter,ine the
reser3e price of a syste, )a*anced and n)a*anced2
Conclu&ion
9n this speci0c sitation% M$S $orporation did not re4est a speci0c decision to )e ,ade2
9nstead% they asked to 0nd the 'reser3e price+ that represents the ,a-i,, price they ere
i**in/ to pay for a certain type of contract2 The reser3e price for a syste, that ses )a*ancin/
as =1#%"2#" and the reser3e price for a syste, ithot )a*ancin/ a,onted to
;B1%="2;B2 :ased so*e*y o these n,)ers% M$S $orporation o*d ha3e to pay ,ore ,oney
for a syste, that ses )a*ancin/ than a syste, that doesn
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Im7lementing t"e @reen Initiatie$e%ning t"e Problem
8o**oin/ the e3er /roin/ trend of /oin/ /reen and the desire to )e in co,p*iance ith ftre/o3ern,ent restrictions% M$S $orporation desired to 'redce their ,onth*y co,)stion
e,issions to *ess than or e4a* to 1" ,etric tons across its faci*ities2+ To do this% they are *ookin/
into prchasin/ so*ar pane*s that o*d ha3e a *ife cyc*e of 03e years and a**o the co,pany to
recei3e a credit of 2 ,etric tons of e,issions ,onth*y for e3ery pane* insta**ed2 The pane*s
o*d )e ot0tted at certain *ocations and o*d prodce ener/y for 1""" Q hors2 Each pane*
cost 1D"""2 9n addition% M$S $orporation can prchase car)on credits to oset e,issions o3er
their thresho*d2 So ith that infor,ation% M$S $orporation o*d *ike to kno ho ,any so*ar
pane*s to prchase and ho ,any car)on credits they sho*d prchase at a speci0c price for
car)on credit2
Anal#&i&
The ana*ysis i** )e done separate*y for the to cases@ )a*ance sp*ittin/ po*icy and opti,a*
sp*ittin/
:a*ance Sp*ittin/ Po*icy
8irst% e ca*c*ated the tota* a,ont of car)on credit needed for each ,onth and the n,)er of
so*ar pane*s and their opti,a* *ocations2 At this sta/e% the price of a car)on credit is nknon so
e te,porari*y ass,ed car)on credit prchase does not e-ist for ca*c*ation prposes2
e se the sa,e ta)*e that e sed for pro)*e, 1>re,o3a* of po*icy?2 To this ta)*e% e added
to 3aria)*e co*,ns@ e,issions per prodcts and tota* e,issions2 A so*ar pane* ta)*e is a*so
added to ca*c*ate the cost of so*ar pane*s and ho to *ocate so*ar pane* p*ace,ent so that the
cost is ,ini,a*2 An e-a,p*e of or ca*c*ation >for No3e,)er? is in ta)*e ;a on the fo**oin/
pa/e2
Para,eter e-p*anation@
Tota* E,ission e,ission per prodct I J prodcts2 This /i3es the a,ont of e,ission of
each prodction *ine2 The tota* e,ission of the ho*e co,pany as a*so ca*c*ated as the
s, of a** e,ission2
J so*ar pane*s@ the n,)er of so*ar pane*s are p*aced at a partic*ar factory deter,ined
)y E-ce* So*3er
$ost!,onth@ $ost of ha3in/ an e-tra so*ar pane*2 Each so*ar pane* cost 1D%""" o3er 03e
years hich is an e4i3a*ent of ;""!,onth
Qh!Unit@ each so*ar pane* pro3ides its factory 1""" kh of free ener/y2 The tota* Qh is
the tota* a,ont of free ener/y each factory /ets2
Tota* $ credits@ 2 I J so*ar pane*s2 Tota* car)on credit that a factory /ets for insta**in/
so*ar pane*s2
Ener/y cost sa3e tota* Qh I ener/y cost >Q? per hor2 The a,ont of ,oney each
factory sa3es de to free ener/y pro3ided )y so*ar pane*s2 This has to )e s,a**er than the
s, of prodction cost of that factory2
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E,ission 5 so*ar car)on credits tota* e,ission 5 tota* $ credit2 This n,)er has to )e
*oer than 1" >,etric tons? in order to ,eet the Green 9nitiati3e re4ire,ent2
So*ar $ost J so*ar pane*s I cost!,onth2 The tota* cost of insta**in/ a** so*ar pane*s in
that ,onth2
Tota* cost s, >cost? L so*ar cost 5 s, >ener/y cost sa3e?2 The tota* cost of prodction2
A** other para,eters are e-p*ained pre3ios*y in re,o3a* of po*icy pro)*e,2
E-ce* So*3er is set to ,ini,i.e to tota* tost and retrns tecision taria)*es of tors of each
factory rn and a,ont of so*ar pane*s each factory insta**ed2
The res*ts of this ca*c*ation are s,,ari.ed in the ta)*e )e*o2
J so*ar
pane*s
an 8e) Mar Apr May n * A/ Sep Oct No3 (
Troy% NY " " " " " " " " " " " "
Neark% N = # D ; = ; B " = 1F ;
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Harris)r/%
PA
" " " " " " " " " " " "
$ost so*ar 1D;
1
1F1"
;
1FFC
=
"D=
B
1D##
C
1F1#
#
1D=1
B
1FB;
D
1#=
1;
1F1"
;
BFB
;
;
F$ credit
needed
"2;C
=B
"2C;
DB
121"
#
12=#
##
"2BB
#B
"2CC
=F
"2=1
"F
"2F
"#
" "2C;
DB
;2#B
"C
#
=Ta)*e ;) So*ar pane*s ana*ysis
e rea*i.e that no ,atter ho ,any so*ar pane*s M$S $orporation need% they i** *ocate the, atNeark% N for opti,a* cost2 Hence% e kno that the e-tra cost that each so*ar pane* )rin/s is
;""5 1""">21D? 1"!,onth or 1=="!year >)ecase each so*ar pane*s cost ;"" )t sa3e
1""" Qh at the ener/y cost of 21D!Qh in Neark?2
Ne-t% e do the ana*ysis to he*p M$S decide the n,)er of pane*s to )y dependin/ on price2
The a*/orith,@ e increase price of car)on credit fro, "!car)on credit nti* the e-tra cost of
)yin/ a** car)on credit is e4a* or hi/her than the e-tra cost of )yin/ a** car)on credit and one
so*ar pane*2 e keep repeatin/ the process for n a,ont of so*ar pane*s >i2e2 e increase price of
car)on credit nti* the e-tra cost of )yin/ a** car)on credit and n so*ar pane*s is e4a* or hi/her
than the e-tra cost of )yin/ a** car)on credit and nL1 so*ar pane*s?2 The reason that thisa*/orith, orks is that% as the price of car)on credit increases% or de,and for car)on credit
prchase decreases and or de,and for so*ar pane*s increases2 The ta)*e )e*o shos the
ca*c*ation2 $ase 1 represents the decision and cost hen M$S )ys n so*ar pane*s% case
represents the decision and cost hen M$S )ys nL1 so*ar pane*s2
The E-ce* So*3er i** try to ,a-i,i.e $ credit price fro, " nti* the tota* e-tra cost of case 1
e4a* or )are*y srpasses the cost of case 2 The res*ts )e*o can )e interpreted as if a car)on
credit price is #B=2BB or )e*o it is opti,a* for M$S to )y " so*ar pane*s and on*y prchase
$ar)on credit to ,eet their /reen initiati3e2
Para,eter E-p*anation@
$ credit price@ the price for one car)on credit2 This is the sa,e for )oth cases2 This 3a*e is
increased fro, " nti* the tota* e-tra cost 1 tota* e-tra cost 2
J so*ar pane*s@ the n,)er of so*ar pane*s prchased for each case2
So*ar pane*s e-tra cost J so*ar pane*s I 1==" )ecase each so*ar pane*s cost e-tra
1=="!year2
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$ credit needed@ the tota* a,ont of car)on credit that /oes o3er 1" ,etric ton *i,it and
need to )e co,pensated )y either so*ar pane*s or $ar)on credit prchase
$ credit 5 so*ar credit $ credit needed 5 "2 I J so*ar pane*s2 The a,ont of $ar)on
credits *eft after sin/ so*ar pane*s car)on credit aka the car)on credit needed to )e
prchase
$ credit prchase cost $ credit price I >$ credit 5 so*ar credit?2 The cost each ,onth to
prchase car)on credit2 This cost is not app*ied for ,onths that ha3e their >$ credit 5 so*ar
credit? 3a*e s,a**er or e4a* to "% sch as Septe,)er in this e-a,p*e2
Tota* e-tra cost So*ar pane*s e-tra cost L $ credit prchase cost2
8ro, the ana*ysis% the res*t ta)*e )e*o is created2 The ta)*e shos the ran/e of $ar)on credit
price and the opti,a*n,)er of so*ar pane*s sho*d )e prchased >e2/2 9f the price of car)on
credit is 1""">hich is )eteen FF12C1 and 1B#=2C% then it is opti,a* for M$S to )y e-act*y
for so*ar pane*s?2 hen the price is a)o3e FF=121F% it is si,p*y not )ene0cia* for M$S to )y
any $ar)on credits and they sho*d insta** a** ;= so*ar pane*s2 At prices sch as ;#"" or C""%
there are ,*tip*e opti,a* so*tions for ho ,any so*ar pane*s sho*d )e )o/ht2 9t depends on
the co,panycost of insta**ation% cost of ,aintenance? to
decide2
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$ar)on creditPrice >per car)on
credit?
Opti,a*n,)erof so*arpane*s8ro, To
"#B=2B=B
=B "
#B=2B=B=B
##21C= 1
##21C=
D1=2CB=1
D1=2CB=1
FF12C"CC1 ;
FF12C"CC1
1B#=2CD =
1B#=2CD
"=F2#;; B
"=F2#;; ="" #
="";"D#2#
B= C;"D#2#
B= ;#"" D
;#"" F
;#"" 1"
;#"" 11
;#"" 1
;#"" 1;
;#"" 1=
;#"" 1B
;#"" 1#
;#"" 1C
;#""BC==2"=
=; 1DBC==2"=
=; C"" 1F
C"" "
C"" 1
C""
C"" ;
C"" =
C"" B
C"" #
C"" C
C"" D
C"" F
C"" ;"
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C"" ;1
C"" ;
C""FF=121F
1B ;;
A)o3e FF=121F ;=Ta)*e ;d
Opti,a* Sp*ittin/
e app*y a si,i*ar ,ethod for opti,a* sp*ittin/2 8irst% e ca*c*ated the tota* a,ont of car)on
credit needed for each ,onth and the n,)er of so*ar pane*s and their opti,a* *ocations2 The
res*ts are shon )e*o2
J so*arpane*s
an 8e) Mar Apr May n * A/ Sep Oct No3 (ec
Troy% NY " " " " " " " " " " C
Neark% N " " " " " " " " " " " "
Harris)r/% PA " " " " " " " " " " " "$ar)on$redit " " " " " " " " " "
12;C#
=2;FD
$ost dierenceF#D2
FF ;;""
Ta)*e ;e
e noticed that No3e,)er and (ece,)er are the on*y to ,onths that e-ceed the $ar)on *i,it2
e a*so rea*i.ed that M$S $orporation i** need to *ocate so*ar pane*s at Troy% NY for opti,a*
cost2 Hence% e kno that the e-tra cost that each so*ar pane*s )rin/ is ;""5 1""">21B? 1B"!,onth or 1D""!year >)ecase each so*ar pane*s cost ;"" )t sa3e 1""" Qh at the
ener/y cost of 21B!Qh in Troy?2
Ne-t% e do the ana*ysis to he*p M$S decide on the n,)er of pane*s to )y dependin/ on price2
e app*y the sa,e a*/orith, that e sed for )a*ance sp*ittin/2 Hoe3er% for the opti,a*
sp*ittin/% e add a constraint that the a,ont of ener/y sa3es a*ays has to )e s,a**er than the
a,ont of ener/y sed at that factory2 This constraint did not appear on the ana*ysis for )a*ance
sp*ittin/ )ecase it is a*ays tre for that scenario2
An e-a,p*e of the e-a,p*e is shon on the ne-t pa/e2 The para,eter stays the sa,e e-cept@
So*ar pane*s prchase cost ;#"" I so*ar pane*s2 The cost of prchasin/ so*ar pane*s is
sed instead of e-tra cost >prchase cost5ener/y sa3in/ cost?2
Tota* ener/y sed in Troy@ the ,a-i,, a,ont of ener/y that can )e sa3ed sin/ so*ar
pane*s2
Ener/y sa3in/ cost Min >tota* ener/y sed in Troy I 21B% J so*ar pane*s I 1B"?2
$onstraint that the a,ont of ener/y sa3es a*ays s,a**er than the a,ont of ener/y
sed at that factory2
Tota* cost so*ar pane*s prchase cost L $ credit prchase cost 5 ener/y sa3in/ cost
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$ase 1
$ase
$credi
tprice
Jso*arpane
*s
$credi
tprice
Jso*arpane
*s
(ecision3aria)*e
552:5
1552:
5
So*arpane*s
prchase cost>1
yr?
CB#""
CF"
"Tota*
an 8e) Mar Apr May n * A/ Sep Oct No3 (ec
$ creditneeded
" " " " " " " " " "12;C
#=2;FD
$ credit5 so*ar
credit>case 1?
5=2 5=2 5=2 5=2 5=2 5=2 5=2 5=2 5=2 5=25
2D=
"21FD
$ credit5 so*ar
credit>case ?
5=2= 5=2= 5=2= 5=2= 5=2= 5=2= 5=2= 5=2= 5=2= 5=2=5
;2"=
5"2""
$ creditprchase cost 1
" " " " " " " " " " " B" B"
$ creditprchase cost
" " " " " " " " " " " " "
Tota*Ener/ysed in
Troy
;"#
;"#
#;#1
D;DF
"CD
;"#
1D#B#
=;;;
D11121
=;;;
B=CB"
DF
Ener/ysa3in/
cost>case 1?
;1B"
;1B";1B
";1B"
;"=12C
;1B"
CFD2;
;1B"11#
2C;1B"
;1B"
;1B";B="
C
Ener/ysa3in/
cost>case ?
;;""
;;"";;"
";;""
;"=12C
;;""
CFD2;
;;""11#
2C;;""
;;""
;;"";#CB
C
O)Kecti3e
fnction
552:5
Total co&t
5
Total co&t
===;2;;;;
===;2;;;;
Ta)*e ;f $ar)on Prchase Price 3s2 So*ar pane*s ana*ysis
8ro, the ana*ysis% the res*t ta)*e )e*o is created2 The ta)*e shos that the ran/e of $ar)on
credit price and the opti,a*n,)er of so*ar pane*s sho*d )e prchased2 hen the price is
a)o3e 11;B"2F% it is si,p*y not )ene0cia* for M$S to )y any $ar)on credits and they sho*d
insta** a** so*ar pane*s2 At prices sch as =B"" or FCB"% there are ,*tip*e opti,a* so*tions
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for ho ,any so*ar pane*s sho*d )e )o/ht2 9t depends on the co,pany preferences and!or
n*isted costs >cost of insta**ation% cost of ,aintenance? to decide2
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$ar)on creditPrice >per
car)on credit?
Opti,a*n,)erof so*arpane*s8ro, To
" =B"" "
=B"" 1
=B""
=B"" ;=B"" =
=B"" B
=B""=CDC2
;= #=CDC2
;= F""" C
F"""F###2#
#C DF###2#
#C FCB" F
FCB" 1"
FCB" 11
FCB" 1
FCB" 1;
FCB" 1=
FCB" 1B
FCB" 1#
FCB" 1C
FCB"
1"""D2
;; 1D1"""D2;; 1"B"" 1F
1"B""11"=12
#C "11"=12
#C11;B"2
F 1
A)o3e 11;B"2F
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Conclu&ion
Balance S7litting O7timal S7litting
$ar)on creditPrice >per car)on
credit?
Opti,a*n,)erof so*arpane*s
$ar)on creditPrice >per
car)on credit?
Opti,a*n,)erof so*arpane*s
8ro, To 8ro, To
"#B=2B=B
=B " " =B"" "#B=2B=B
=B##21
C= 1 =B"" 1##21
C=
D1=2CB
=1 =B"" D1=2CB=1
FF12C"CC1 ; =B"" ;
FF12C"CC1
1B#=2CD = =B"" =
1B#=2CD
"=F2#;; B =B"" B
"=F2#;; ="" # =B""
=CDC2;= #
="";"D#2#
B= C=CDC2
;= F""" C
;"D#2#B= ;#"" D F"""
F###2##C D
;#"" FF###2#
#C FCB" F
;#"" 1" FCB" 1"
;#"" 11 FCB" 11
;#"" 1 FCB" 1
;#"" 1; FCB" 1;
;#"" 1= FCB" 1=
;#"" 1B FCB" 1B
;#"" 1# FCB" 1#;#"" 1C FCB" 1C
;#""BC==2"=
=; 1D FCB"1"""D2
;; 1DBC==2"=
=; C"" 1F1"""D2
;; 1"B"" 1F
C"" " 1"B""11"=12
#C "
C"" 111"=12
#C11;B"2
F 1
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C"" A)o3e 11;B"2F
C"" ;
C"" =
C"" B
C"" #
C"" C
C"" D
C"" FC"" ;"
C"" ;1
C"" ;
C""FF=121F
1B ;;
A)o3e FF=121F ;=Ta)*e ;h
After a thoro/h ana*ysis of the data% it as deter,ined that the decision to )y ,ore car)on
credit o*d ha3e to )e ,ade dependin/ on the crrent sitation at the ti,e2 ith that )ein/said% the research shoed that price ran/es co*d )e ,ade >reference car)on credit price ran/es
in ta)*e ;h? and then co*d dictate ho ,any so*ar pane*s the co,pany sho*d insta**2
Referencin/ ta)*e ;h% the co,pany co*d *ook at the price of the crrent car)on credit and then
insta** the opti,a* choice of so*ar pane*s% fo**oed )y prchasin/ ,ore car)on credits if need )e2
This so*tion to the pro)*e, o*d a**o the M$S $orporation the a)i*ity to esti,ate ho ,any
car)on credits and so*ar pane*s to )y in the ftre2