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7/23/2019 1. Break Even Analysis
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BREAK EVENANALYSIS
Topic 1
Engr. Tafalla
November 30 !01"
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Definition:
ENGINEERING ECONOMY is a discipline
concerned with the systematic evaluation of the costsand enefits of the proposed usiness pro!ects andventures" Its o!ective is to choose which amon# thealternative course of action will #ive the ma$imumenefit at the least cost"
En#ineerin# Economy% therefore involves theapplication of definite laws of Economics% theories ofinvestment and usiness practices to en#ineerin#prolems involvin# cost" It also involves the study ofcost features and other financial data and theirapplications in the field of en#ineerin# as asis fordecision"
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COST CONCEPTS
DEMAND – is the &uantity of a certain commodity that is ou#ht at a certainprice at a #iven place and time"
SUPPLY ' is the &uantity of a certain commodity that is offered for sale at acertain price at a #iven place and time"
FIXED COST ' are costs that do not vary in proportion to the &uantity ofoutput"
VARIABLE COST ' are costs that vary in proportion to &uantity of output"
BREAK EVEN POINT ' is the level of production at which revenue is e$actly
e&ual to total costs
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Elements of Cost:
()Materials
a) Direct Materials are those which are used in thefinished product itself"
) Indirect Materials are those materials used inproduction ut which do not #o into the finishedproduct"
*) +aor
a) Direct +aor is the actual wor, applied directly tothe manufacture of the product
) Indirect +aor is the wor, necessary for theoperation of the factory% ut which cannot eidentified with one particular process or productmanufactured"
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-) Overhead E$penses
E$penses which cannot e allocated to directmaterials or direct lair"
.RIME CO/0 1 Direct Materials Cost 2 Direct +aorCost
.ROD3C0ION CO/0 1 Direct Materials Cost 2 Direct
+aor Cost 2 Overhead CostOr
.ROD3C0ION CO/0 1 .rime Cost 2 Overhead Cost
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LAW OF SUPPLY 0he supply of the commodity varies directly as theprice of the commodity% thou#h not proportionately
/upply
pr ice
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LAW OF DEMAND
0he demand for a commodity varies inversely as the
price of the commodity% thou#h not proportionately
Demand
pr ic
e
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LAW OF DEMAND AND SUPPLY 3nder conditions of perfect competition% the price atwhich any #iven product will e supplied and purchasedis the price that will result in the supply and the demandein# e&ual"
4uantity
pr ic
e
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0he relationship etween price and demand can ee$pressed as a line
5here a is the intercept on the price 6p)a$is and –b isthe slope"
p = a - bD
pr ice
Demand 6D)
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TOTAL REVENUE – VOLUME
RELATIONSHIP
7olume 6D)
Peak point – represents te
Ma!i"#" re$en#e0
O0 8+
Rev
enue
D%
De"an& tat "a!i"i'es
Tota( Re$en#e
2
)(
bDaDTR
or
DbDaTR
pDTR
−=
−=
=
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COST - VOLUME RELATIONSHIP
TFC vcDTC
TFC TVC TC
+=
+=0otal Cost
9i$ed Cost
7ariale CostC
ost
7olume 6D)
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COM)INATION O* COST - VOLUME +
REVENUE VOLUME RELATIONSHIP
b
vca D 2
* −=
Vo(#"e ,D
Represents te
Ma!i"#" Pro.itC
o
s
t
R
e
$
en
#
e
D /
De"an& tat "a!i"i'es
Tota( Pro.it
or Tota( Cost
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9ormulas:
.rice:
0otal Revenue:
0otal Cost :
bDa p −=
2
)(
bDaDTR
or
DbDaTR
pDTR
−=
−=
=
TFC vcDTC
TFC TVC TC
+=
+=
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.rofit:
Demand that ma$imies Revenue
Demand that ma$imies .rofit
6Optimum .rofit)
TFC DvcabD P
TFC vcD DbDa P
TFC vcD pD P
TC TR P
−−+−=
−−−=
+−=
−=
)(
)(
)(
2
b
a D 2=
b
vca
D 2
* −=
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;rea, even points: .rofit 1 <
TFC DvcabD
TFC DvcabD P
TFC vcD DbDa P
TFC vcD pD P
TC TR P
−−+−=
−−+−=
−−−=
+−=
−=
)(0
)(
)(
)(
2
2
)(2
))((4)()( 2
'
b
TFC bvcavca D
−
−−−−±−−=
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I" CO/0 CONCE.0/
;: .rice is not constant
;rea, even point:
+ o s s
. R O 9 I 0
7olume 6D)
Revenue
CO/0
or
)reak E$en Point
0ere TR=TC
vc p
TFC D
−=
' vc p
TFC D
−=
'
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E$amples:(" 8 company produces circuit oards to update the
outdated computer e&uipment" 0he fi$ed cost is =>*%<<<per month and the variale cost is =?- per circuit oard"
0he sellin# price per unit is p 1 =(?< ' <"<*D" Ma$imumoutput of the plant is ><<< units per month"
6a) Determine the optimum value for this product"
6) 5hat is the ma$imum profit per month@
6c) 8t what volumes does rea,Aeven occur@
6d) 5hat is the companyBs ran#e of profitale demand@
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E$amples*" 8 lar#e semiconductor plant has appro$imately ? of sales
due to a sin#le circuit desi#n" 0he plant can therefore econsidered to produce -%<<<%<<< printed circuit oards 6.C;s)per year" .resently% the plant is operatin# at < of capacity"0he sellin# price of the .C; is p 1 =("*? ' 6(<A E )D% and the
variale cost per .C; is =(?"F?" 8t ero output% the plantsannual fi$ed costs are =(%<<<%<<< and are appro$imatelyconstant up to the ma$imum production &uantity per year"
a" 5hat is the present e$pected annual profit or loss 6<capacity)@
" 5hat the percentage of production capacity that will result inoptimal operation@ 5hat is the ma$imum profit or minimum lossat this optimal volume@
"Determine at what demand6s) rea,even occurs in theoperation
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E$amples:12 8 manufacturin# company leases for =(<<%<<< per year auildin# that houses its manufacturin# facilities" Inaddition% the machinery in the uildin# is ein# paid for
installments of =*<%<<< per year" Each unit of productproduced costs =(? in laor and =(< in materials and cane sold for =><"
a"ow many units per year must e sold for the companyto rea, even@
" If (<%<<< units per year are sold% what is the annualprofit@
c" If the sellin# price is lowered to =-? per unit% how manyunits must e sold each year for the company to earn aprofit of =<%<<< per year@
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>"8 company produces and sells a consumer product and thusfar has een ale to control the volume of the product yvaryin# the sellin# price" 0he company is see,in# to ma$imie
its net proHt" It has een concluded that the relationshipetween price and demand% per month% is appro$imately
where p is the price per unit in dollars" 0he H$ed cost is=(%<<< per month% and the variale cost is =*< per unit"
Otain the answer mathematically to the followin# &uestions:a" 5hat is demand that will ma$imie revenue per monthand the ma$imum revenue
" 5hat is the optimal numer of units that should e
produced and sold per month@c" 5hat is the ma$imum proHt per month@
d" 5hat are the rea,even sales &uantities and the ran#eof proHtale demand 6volume)@
D = p500 5 ,−
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?" 8 plant operation has fi$ed cost of =*%<<<%<<< peryear% and its output capacity is (<<%<<< electricalappliances per year" 0he variale cost is =>< per unit%and the product sells for =< per unit"
a) 5hat is the annual rea, even volume of thisproduct@
) Compare annual profit when the plant is operatin# at< capacity with the plant operation at (<<
capacity" 8ssume that the first < of capacity outputis sold at =< per unit and that the remainin# (< ofproduction is sold at =F< per unit"
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E$amples" 8 company has estalished that the relationship
etween the sales price for one of its products andthe &uantity sold per month is appro$imately D 1 F< ' (<p units" 0he fi$ed cost is =<< per month% andthe variale cost is =-< per unit produced" 5hatnumer of units should e produced per month andsold to ma$imie net profit@ 5hat is the ma$imum
profit per month@ Determine the ran#e of profitaledemand"
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E$amples:F" Te ann#a( .i!e& 3osts .or a p(ant are P4556555
an& te $ariab(e 3osts are P4756555 at
859#ti(i'ation o. a$ai(ab(e 3apa3it: 0it net sa(es
o. P;<565552 at is te break e$en point in #nits
o. protion i. te se((in> pri3e per #nit is P752
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" /uppose we ,now that p1(%<<< ' DJ?% where p 1
price in dollars and D 1 annual demand" 0he totalcost per year can e appro$imated y =(%<<< 2 *D* "
a"Determine the value of D that ma$imies profit"
"/how that in part6a) profit has een ma$imied ratherthan minimied"
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Ans0er?
,a p 1 (%<<< A <"*D0C 1 (%<<< 2 *D*
.rofit 1 0otal Revenue A 0otal Cost 1 6(%<<< A <"*D)D A 6(%<<< 2 *D*)1 (%<<<D A *"*D* A (%<<<
DK 1 **F"*F units per year
,b
/ince the second derivative is ne#ative% profit has een ma$imied atDK"
d
d
Profit
D = 1,000 - 4.4 D = 0*
d
d
2 (Profit)
D = - 4.4 < 0
2
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" 0he fi$ed cost for a steam line per meter of pipe is
=>?<L 2 =?< per year" 0he cost for loss of heat fromthe pipe per meter is =>"JL(J* per year" ere Lrepresents the thic,ness of insulation in meters andL is a continuous desi#n variale"
a" 5hat is the optimum thic,ness of the insulation@" ow do you ,now that your answer in 6a)minimies total cost per year@
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Ans0er?
,a 0otal 8nnual Cost 608C) 1 9i$ed cost 2 Cost of eat +oss
1 >?<L 2 ?< 2
LK 1 <"<-<? meters,b for L <"
/ince the second derivative is positive% L K 1 <"<-<? meters is a minimum cost thic,ness"
,3 0he cost of the e$tra insulation 6a directly varyin# cost) is ein# tradedAoff a#ainst thevalue of reduction in lost heat 6an indirectly varyin# cost)"
3/2X
2.40 -450=0=
X
(TAC)
d
d
0.00533=450
2.40
=X3/2
d
d
2 (TAC)
X =
3.6
X > 0
2 5/2
4 80.
X1/ 2
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F" 8 local defense contractor is considerin# the
production of firewor,s as a way to reducedependence on the military" 0he variale cost per unitis =><D" 0he fi$ed cost that can e allocated to theproduction of firewor,s is ne#li#ile" 0he pricechan#ed per unit will e determined y the e&uationp1=(<A6?)D% where D represents demand in unitssold per wee,"
a"5hat is the optimum numer of units the defensecontractor should produce in order to ma$imie profit
per wee,@"5hat is the profit if the optimum numer of units are
produced@
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,a 0otal Revenue 1 p D1 6(< ' ?D)D 1 (<D ' ?D*
0otal Cost 1 6><D)D 1 ><D*
0otal .rofit 1 A?D* 2 (<D ' ><D*
1 A (<D 2 (< ' <D 1 <
<D 1 (< DK * unitsJwee,
1 A< < ∴ ma$imum profit
6) 0otal .rofit 1 A?6**) 2 (<6*) ' ><6**) 1 A*< 2 -< ' (< 1 =(< J wee,
D
(Profit)
d
d
d
d
2(Profit)
D2
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/eatwor,:(" 8 company has determined that the price and that
monthly demand of one of its products are related ythe e&uation
0he associated fi$ed costs are =(%(*?Jmonth% and thevariale costs are =(<<Junit"
a" 5hat is the optimal numer of units to ma$imie
revenue and the ma$imum revenue@" 5hat is the optimal numer of units that should eproduced and sold each month to ma$imie profit@
c" 5hat are the rea, even points@
)400( p D −=
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*" 8 plant operation has fi$ed cost of =*%<<<%<<< peryear% and its output capacity is (<<%<<< electricalappliances per year" 0he variale cost is =>< per unit%and the product sells for =< per unit"
a) 5hat is the annual rea, even volume of thisproduct@
) Compare annual profit when the plant is operatin#at < capacity with the plant operation at (<<
capacity" 8ssume that the first < of capacity outputis sold at =< per unit and that the remainin# (< ofproduction is sold at =F< per unit"
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-" 8 manufacturer is currently sellin# (<<< decorative lamps
a month to the retailers at a price of .<< per lamp" Itsestimates that for each .?< increase in the price will sell*< fewer lamps each month" 0he manufacturerBs costconsists of a fi$ed overhead of .-<<%<<<J month plus.-<< per lamp for laor and materials"
a"/et up the total cost function
"/et up the demand function
c" 9ind the ;rea, Even points
d"9ind the volume that will ma$imie profite"5hat is the ma$imum profit@
f" 5hat is the volume of sales that will ma$imie your salesrevenue@
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>" ;ra## P /tratton Company manufactures a specialiedmotor for chain saws" 0he company e$pects to manufactureand sell -<%<<< motors in year *<<(" It can manufacture an
additional (<%<<< motors without addin# new machineryand e&uipment" Its pro!ected total costs for the -<%<<< unitsare as follows:Direct Materials =(?<%<<<Direct +aor (<<%<<<Manufacturin# Overhead:
7ariale .ortion (<<%<<<9i$ed .ortion <%<<<
/ellin# and 8dministrative costs:7ariale .ortion (<%<<<9i$ed .ortion F<%<<<
0he sellin# price for the motor is =<"a" 5hat is the total manufacturin# cost per unit if -<<%<<<motors are produced@" 5hat is the total manufacturin# cost per unit if ><%<<<motors are produced@c" 5hat is the rea, even price on the motors@
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5)(0.08)(1.
5.88
D(T!)d
d
5)(0.08)(1.
5.88
D
(TP)
d
d
5)(0.08)(1.
5.48
5)(0.08)(1.
5.48
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,3 9C 1 Q=-?<%<<< A <"(6=-?<%<<<)6(*) 1
=-%F<%<<< per year 6(< decrease)
vc 1 Q=<"?< 2 <"( 6=<"?<) 1 =<"?? per unit ofsales 6(< increase)
0hus% no chan#e occurred in the ori#inalrea,even point"
′D ="3,80,000
"1 - "0.55 = "8,400,000 #$r %$&r
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;-;5 ,a
,b .rofit6+oss) 1 0otal Revenue A 0otal Cost
6< Capacity) 1 <%<<< 6=<) A Q=*%<<<%<<< 2<%<<< 6=><)
1 =*%?<<%<<< per year
6(<< Capacity) 1 Q<%<<<6=<) 2 (<%<<<6=F<) A
Q=*%<<<%<<< 2 (<<%<<<6=><) 1 =*%<<%<<< per year
′D =C
# - ' =
"2,000,000
("0 - "40) / it = 40,000 it+ #$r %$&r
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)REA@ – EVEN ANALSIS6 TO ALTERNATIVES
In&#str: is .a3e& 0it 3ertain sit#ations 0ere t0o or "ore a(ternati$es 3an be3onsi&ere&2 en te 3ost .or t0o a(ternati$es is a..e3te& b: a 3o""on &e3ision$ariab(e6 tere "a: e!ist a $a(#e o. te $ariab(e .or 0i3 te t0o a(ternati$es 0i((in3#r eB#a( 3ost2 Tis $a(#e is kno0n as te break-e$en 3ost2 )e(o0 tis 3ost6 one"eto& 0i(( be "ore e3ono"i3a(6 an& abo$e tis 3ost6 te oter 0i(( pro$e to bebetter e3ono"i3a((:2
0C8
0C; 0otal Cost
DB 7olume 6D) 0C8 1 0C;
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E$amples42 T0o "an#.a3t#rin> "eto&s are bein>
3onsi&ere&2 Meto& A as a .i!e& 3ost o.
P6555 an& a $ariab(e 3ost o. P52 Meto& )
as a .i!e& 3ost o. P;55 an& a $ariab(e 3ost o.452 *or 0at protion $o(#"e 0o#(& one
pre.er ,a Meto& A6 an& ,b Meto& )
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;2 T0o 3o"panies are en>a>e& in te "an#.a3t#re o. sirts2Co"pan: A6 #sin> "ost(: an&0ork6 as a .i!e& 3ost "ont(:e!pense o. P76555 an& a $ariab(e 3ost o. P4255 per sirt2Co"pan: ) as been ab(e to "e3ani'e "ost o. its
operations6 an& it .in&s its .i!e& "ont(: e!penses areP<56555 an& te $ariab(e 3ost per sirt is P4;252
a2 Ho0 "an: sirts so#(& be "an#.a3t#re& b: ea3"ont so tat te tota( 3ost 0i(( be te sa"e .or te t0o3o"panies
b2 I. ea3 sirt se((s .or P1;255 to te retai(ers6&eter"ine te "ont(: >ross pro.it .or ea3 3o"pan:2