The document discusses including passive fire protection elements in a new manufacturing building even though local codes only require an automatic sprinkler system. Some factors to consider in a cost-benefit analysis of including passive elements are: 1) Direct costs of installing passive elements like fire-rated walls and doors. 2) Indirect costs like potential reductions in insurance premiums from additional fire protection. 3) Benefits in the form of reduced risk of property damage and business interruption from fires.

1. one engineering question
engineering writing question and need an explanation and answer to help me learn.
hello
please see the question below and answer it perfectly. use some sources related to fire and
see the attachment for reference.
I NEED IT IN 20 MINUTES. it should not be long. half a page is fine. thanks
You are designing a building to house your new manufacturing line. The local codes require
the building to be sprinklered while providing very little passive protection features.
Obviously, you have to provide a sprinkler system, but would you consider including
passive elements in the structure as well – even though they are not required? Without
attempting to assign costs, what items would you consider in the cost-benefit analysis?
Include both direct and indirect costs.
Requirements: as required
5–93IntroductionEngineering economics is the application of economictechniques to the
evaluation of design and engineering al-ternatives.1The role of engineering economics is to
assessthe appropriateness of a given project, estimate its value,and justify it from an
engineering standpoint.This chapter discusses the time value of money andother cash-flow
concepts, such as compound and contin-uous interest. It continues with economic practices
andtechniques used to evaluate and optimize decisions on se-lection of fire safety strategies.
The final section expandson the principles of benefit-cost analysis.An in-depth treatment of
the practices and techniquescovered in this compilation is available in the
ASTMcompilation of standards on building economics.2TheASTM compilation also includes
case illustrations show-ing how to apply the practices and techniques to invest-ment
decisions.Abroader perspective on the application of engin-eering economics to fire
protection engineering can befound in The Economics of Fire Protection by Ramachan-
dran.3This work is intended as a textbook for fire protec-tion engineers and includes
material and references thatexpand on several other chapters of this section of theSFPE
handbook.Cash-Flow ConceptsCash flow is the stream of monetary (dollar) values—costs
(inputs) and benefits (outputs)—resulting from aproject investment.Time Value of
MoneyThe following are reasons why $1000 today is“worth” more than $1000 one year
from today:1.Inflation2.Risk3.Cost of moneyOf these, the cost of money is the most
predictable,and, hence, it is the essential component of economicanalysis. Cost of money is
represented by (1)money paidfor the use of borrowed money, or (2)return on invest-ment.

2. Cost of money is determined by an interest rate.Time value of money is defined as the time-
dependentvalue of money stemming both from changes in the pur-chasing power of money
(inflation or deflation) and fromthe real earning potential of alternative investments
overtime.Cash-Flow DiagramsIt is difficult to solve a problem if you cannot see it.
Theeasiest way to approach problems in economic analysis isto draw a picture. The picture
should show three things:1.Atime interval divided into an appropriate number ofequal
periods2.All cash outflows (deposits, expenditures, etc.) in eachperiod3.All cash inflows
(withdrawals, income, etc.) for eachperiodUnless otherwise indicated, all such cash flows
are con-sidered to occur at the end of their respective periods.Figure 5-7.1 is a cash-flow
diagram showing an out-flow or disbursement of $1000 at the beginning of year 1and an
inflow or return of $2000 at the end of year 5.NotationTo simplify the subject of economic
analysis, sym-bolsare introduced to represent types of cash flows andSECTION
FIVECHAPTER 7Engineering EconomicsJohn M.Watts, Jr., and Robert E.ChapmanDr. John M.
Watts, Jr., holds degrees in fire protection engineering,industrial engineering, and
operations research. He is director of theFire Safety Institute, a not-for-profit information,
research, and edu-cational corporation located in Middlebury, Vermont. Dr. Watts
alsoserves as editor of NFPA’s Fire Technology. Dr. Robert E. Chapman is an economist in
the Office of Applied Eco-nomics, Building and Fire Research Laboratory, National Institute
ofStandards and Technology.05-07.QXD 11/14/2001 11:53 AM Page 93
interest factors. The symbols used in this chapter conformto ANSI Z94;4however, not all
practitioners follow thisstandard convention, and care must be taken to avoidconfusion
when reading the literature. The followingsymbols will be used here: PCPresent sum of
money ($)FCFuture sum of money ($)NCNumber of interest periodsiCInterest rate per
period (%)Acomplete list of the ANSI Z94 symbols is given in Ap-pendix Ato this
chapter.Interest CalculationsInterest is the money paid for the use of borrowedmoney or
the return on invested capital. The economiccost of construction, installation, ownership, or
operationcan be estimated correctly only by including a factor forthe economic cost of
money.Simple interest:To illustrate the basic concepts of inter-est, an additional notation
will be used:F(N)CFuture sum of money after NperiodsThen, for simple
interest,F(1)CP=(P)(i)CP(1=i)andF(N)CP=(N)(P)(i)CP(1=Ni)For example: $100 at 10
percent per year for 5 yr yieldsF(5)C100[1=(5)(0.1)]C100(1.5)C$150However, interest is
almost universally compounded toinclude interest on the interest.Compound
interestF(1)CP=(P)(i)CP(1=i)is the same as simple interest,F(2)CF(1)=F(1)(i)Interest is
applied to the new sum:C(F)(1)(1=i)CP(1=i)2F(3)CF(2)(1=i)CP(1=i)3and by mathematical
induction,F(N)CP(1=i)NEXAMPLE:$100 at 10 percent per year for 5 yr
yieldsF(5)C100(1=0.1)5C100(1.1)5C100(1.61051)C$161.05which is over 7 percent greater
than with simple interest.EXAMPLE:In 1626 Willem Verhulst bought Manhattan Islandfrom
the Canarsie Indians for 60 florins ($24) worth ofmerchandise (a price of about 0.5 cents
per hectare [0.2cents per acre]). At an average interest rate of 6 percent,what is the present
value (2001) of the Canarsies’ $24?FCP(1=i)NC$24(1=0.06)375C$7.4?1010Seventy-four
billion dollars is a reasonable approxima-tion of the present land value of the island of
Manhattan.Interest FactorsInterest factors are multiplicative numbers calculatedfrom
interest formulas for given interest rates and peri-ods. They are used to convert cash flows

3. occurring atdif-ferent times to a common time. The functional formatsused to represent
these factors are taken from ANSI Z94,and they are summarized in Appendix B to this
chapter.Compound Amount FactorIn the formula for finding the future value of a sum
ofmoney with compound interest, the mathematical expres-sion (1=i)Nis referred to as the
compound amount factor,represented by the functional format
(F/P,i,N).Thus,FCP(F/P,i,N)Interest tables:Values of the compound amount, pre-sent worth,
and other factors that will be discussedshortly, are tabulated for a variety of interest rates
andnumber of periods in most texts on engineering economy.Example tables are presented
in Appendix C to this chap-ter. Although calculators and computers have greatly re-duced
the need for such tables, they are often still usefulfor interpolations.5–94Fire Risk
Analysis1$1000$200034502Figure 5-7.1.Cash-flow diagram.05-07.QXD 11/14/2001 11:53
AM Page 94
Present WorthPresent worth is the value found by discounting fu-ture cash flows to the
present or base time.Discounting:The inverse of compounding is determin-ing a present
amount which will yield a specified futuresum. This process is referred to as discounting.
The equa-tion for discounting is found readily by using the com-pounding equation to solve
for P in terms of F:PCF(1=i)>NEXAMPLE:What present sum will yield $1000 in 5 yr at 10
percent?PC1000(1.1)>5C1000(0.62092)C$620.92This result means that $620.92
“deposited” today at 10percent compounded annually will yield $1000 in 5 yr.Present
worth factor:In the discounting equation, theexpression (1=i)>Nis called the present worth
factorand isrepresented by the symbol (P/F,i,N).Thus, for the presentworth of a future sum
at i percent interest for N periods,PCF(P/F,i,N)Note that the present worth factor is the
reciprocal of thecompound amount factor. Also note
that(P/F,i,N)C1(F/P,i,N)EXAMPLE:What interest rate is required to triple $1000 in 10
years?PCF3C(P/F,i,10)therefore,(P/F,i,10)C13From Appendix
C,(P/F,10%,10)C0.3855and(P/F,12%,10)C0.3220By linear interpolation,iC11.6%Interest
PeriodsNormally, but not always, the interest period is takenas 1 yr. There may be
subperiods of quarters, months,weeks, and so forth.Nominal versus effective interest:It is
generally as-sumed that interest is compounded annually. However,interest may be
compounded more frequently. When thisoccurs, there is a nominal interestor annual
percentage rateand an effective interest,which is the figure used in calcu-lations. For
example, a savings bank may offer 5 percentinterest compounded quarterly, which is not
the same as5 percent per year. Anominal rate of 5 percent com-pounded quarterly is the
same as 1.25 percent every threemonths or an effective rate of 5.1 percent per year.
IfrCNominal interest rate,andMCNumber of subperiods per yearthen the effective interest
rate isiC‹1=rMM>1EXAMPLE:Credit cards usually charge interest at a rate of 1.5percent per
month. This amount is a nominal rate of 18percent. What is the effective
rate?iC(1=0.015)12>1C1.1956>1C19.56%Continuous interest:Aspecial case of effective
interestoccurs when the number of periods per year is infinite.This represents a situation of
continuous interest,also re-ferred to as continuous compounding. Formulas for con-tinuous
interest can be derived by examining limits as Mapproaches infinity. Formulas for interest
factors usingcontinuous compounding are included in Appendix B.Continuous interest is
compared to monthly interest inTable 5-7.1.EXAMPLE:Compare the future amounts

4. obtained under variouscompounding periods at a nominal interest rate of 12 per-cent for 5
yr, if PC$10,000.(See Table 5-7.2.)Series PaymentsLife would be simpler if all financial
transactions werein single lump-sum payments, now or at some timein theEngineering
Economics5–95EffectiveNominal
%MonthlyContinuous55.15.11010.510.51516.116.22021.922.1Table 5-7.1Continuous
Interest (%)05-07.QXD 11/14/2001 11:53 AM Page 95
future. However, most situations involve a series of regu-lar payments, for example, car
loans and mortgages.Series compound amount factor:Given a series of regu-lar payments,
what will they be worth at some future time?LetACthe amount of a regular end-of-period
paymentThen, note that each payment, A, is compounded for adifferent period of time. The
first payment will be com-pounded for N>1periods (yr):FCA(1=i)N>1and the second
payment for N>2periods:FCA(1=i)N>2and so forth. Thus, the total future value
isFCA(1=i)N>1=A(1=i)N>2=ß=A(1=i)=AorFCA[(1=i)N>1]iThe interest expression in this
equation is known as theseries compound amount factor,(F/A,i,N),thusFCA(F/A,i,N)Sinking
fund factor:The process corresponding to theinverse of series compounding is referred to as
a sinkingfund;that is, what size regular series payments are neces-sary to acquire a given
future amount?Solving the series compound amount equation for A,ACF48i[(1=i)N>1]Or,
using the symbol (A/F,i,N)for the sinking fund factorACF(A/F,i,N)Here, note that the sinking
fund factor is the reciprocal ofthe series compound amount factor, that is,
(A/F,i,N)C1/(F/A,i,N).Capital recovery factor:It is also important to be able torelate regular
periodic payments to their present worth;for example, what monthly installments will pay
for a$10,000 car in 3 yr at 15 percent?Substituting the compounding equation F
CP(F/P,i,N)in the sinking fund equation, ACF(A/F,i,N),yieldsACP(F/P,i,N)(A/F,i,N)And,
substituting the corresponding interest factors givesACP[i(1=i)N][(1=i)N>1]In this
equation, the interest expression is known as thecapital recovery factor,since the equation
defines a regularincome necessary to recover a capital investment. Thesymbolic equation
isACP(A/P,i,N)Series present worth factor:As with the other factors,there is a
corresponding inverse to the capital recoveryfactor. The series present worthfactor is found
by solvingthe capital recovery equation for P.PCA[(1=i)N>1][i(1=i)N]or,
symbolicallyPCA(P/A,i,N)Other Interest Calculation ConceptsAdditional concepts involved
in interest calculationsinclude continuous cash flow, capitalized costs, begin-ning of period
payments, and gradients.Continuous cash flow:Perhaps the most useful func-tion of
continuous interest is its application to situationswhere the flow of money is of a
continuous nature. Con-tinuous cash flowis representative for1.Aseries of regular payments
for which the interval be-tween payments is very short2.Adisbursement at some unknown
time (which is thenconsidered to be spread out over the economic period)5–96Fire Risk
AnalysisCompoundingAnnualSemi-
annualQuarterlyMonthlyWeeklyDailyHourlyInstantaneouslyM12412523658760ãi12.00012
.36012.55112.68312.73412.74712.74912.750NM5102060260182543,800ãF/P1.762341.7
90851.806111.816701.8208601.8219381.8220611.822119aF17,623.4017,908.5018,061.1
018,167.0018,208.6018,219.3818,220.6118,221.19Table 5-7.2Example of Continuous
Interest N C5 yr,r C12%aF/P (instantaneous) CeNiCe5(0.12)Ce0.6.05-07.QXD 11/14/2001
11:53 AM Page 96

5. Factors for calculating present or future worth of aseries of annual amounts, representing
the total of a con-tinuous cash flow throughout the year, may be derivedbyintegrating
corresponding continuous interest factorsover the number of years the flow is
maintained.Continuous cash flow is an appropriate way to han-dle economic evaluations of
risk, for example, the presentvalue of an annual expected loss.Formulas for interest factors
representing continu-ous, uniform cash flows are included in Appendix B.Capitalized
costs:Sometimes there are considerations,such as some public works projects, which are
consideredto last indefinitely and thereby provide perpetual service.For example, how
much should a community be willingto invest in a reservoir which will reduce fire
insurancecosts by some annual amount, A? Taking the limit of theseries present worth
factor as the number of periods goesto infinity gives the reciprocal of the interest rate.
Thus,capitalized costsare just the annual amount divided by theinterest rate. When
expressed as an amount required toproduce a fixed yield in perpetuity, it is sometimes re-
ferred to as an annuity.Beginning-of-period payments:Most returns on invest-ment (cash
inflows) occur at the end of the period duringwhich they accrued. For example, a bank
computes andpays interest at the end of the interest period. Accordingly,interest tables,
such as those in Appendix C, are computedfor end-of-year payments. For example, the
values of thecapital recovery factor (A/P,i,N)assume that the regularpayments, A, occur at
the end of each period.On the other hand, most disbursements (cash out-flows) occur at the
beginning of the period (e.g., insurancepremiums). When dealing with beginning-of-period
pay-ments, it is necessary to make adjustments. One methodof adjustment for beginning-of-
period payments is to cal-culate a separate set of factors. Another way is to
logicallyinterpret the effect of beginning-of-period payments for aparticular problem, for
example, treating the first pay-ment as a present value. The important thing is to recog-nize
that such variations can affect economic analysis.Gradients:It occasionally becomes
necessary to treatthe case of a cash flow which regularly increases or de-creases at each
period. Such patterned changes in cashflow are called gradients.They may be a constant
amount(linear or arithmetic progression), or they may be a con-stant percentage
(exponential or geometric progression).Various equations for dealing with gradient series
may befound in Appendix B.Comparison of AlternativesMost decisions are based on
economic criteria. In-vestments are unattractive, unless it seems likely they willbe
recovered with interest. Economic decisions can be di-vided into two classes:1.Income-
expansion—that is, the objective of capitalism2.Cost-reduction—the basis of
profitabilityFire protection engineering economic analysis is pri-marily concerned with
cost-reduction decisions, findingthe least expensive way to fulfill certain requirements,
orminimizing the sum of expected fire losses plus invest-ment in fire protection.There are
four common methods of comparing alter-native investments: (1)present worth, (2)annual
cost,(3)rate of return, and (4)benefit-cost analysis. Each ofthese is dependent on a selected
interest rate or discountrate to adjust cash flows at different points in time.Discount
RateThe term discount rateis often used for the interestrate when comparing alternative
projects or strategies.Selection of discount rate:If costs and benefits accrueequally over the
life of a project or strategy, the selectionof discount rate will have little impact on the
estimatedbenefit-cost ratios. However, most benefits and costs oc-cur at different times

6. over the project life cycle. Thus,costs of constructing a fire-resistive building will be in-
curred early in contrast to benefits, which will accrue overthe life of the building. The
discount rate then has a sig-nificant impact on measures such as benefit-cost ratios,since
the higher the discount rate, the lower the presentvalue of future benefits.In view of the
uncertainty concerning appropriate dis-count rate, analysts frequently use a range of
discountrates. This procedure indicates the sensitivity of the analy-sis to variations in the
discount rate. In some instances,project rankings based on present values may be
affectedby the discount rate as shown in Figure 5-7.2. Project Aispreferred to project B for
discount rates below 15 percent,while the converse is true for discount rates greater than15
percent. In this instance, the decision to adopt project Ain preference to project B will
reflect the belief that the ap-propriate discount rate is less than or equal to 15
percent.Engineering Economics5–9702015105Discount rate (%)Net present valueBAFigure
5-7.2.Impact of discount rate on project selection.05-07.QXD 11/14/2001 11:53 AM Page
97
Acomparison of benefits and costs may also be usedto determine the payback period for a
particular project orstrategy. However, it is important to discount future costsor benefits in
such analyses. For example, an analysis ofthe Beverly Hills Supper Club fire compared
annual sav-ings from a reduction in insurance premiums to the costsof sprinkler
installation. Annual savings were estimated at$11,000, while costs of sprinkler installation
ranged from$42,000 to $68,000. It was concluded that the installationwould have been paid
back in four to seven years (de-pending on the cost of the sprinklers). However, thisanalysis
did not discount future benefits, so that $11,000received at the end of four years was
deemed equivalentto $11,000 received in the first year. Once future benefitsare discounted,
the payback period ranges from five toeleven years with a discount rate of 10
percent.Inflation and the discount rate:Provision for inflationmay be made in two ways:
(1)estimate all future costsand benefits in constant prices, and use a discount ratewhich
represents the opportunity cost of capital in the ab-sence of inflation; or (2)estimate all
future benefits andcosts in current or inflated prices, and use a discount ratewhich includes
an allowance for inflation. The discountrate in the first instance may be considered the real
dis-count rate, while the discount rate in the second instanceis the nominal discount rate.
The use of current or inflatedprices with the real discount rate, or constant prices withthe
nominal discount rate, will result in serious distor-tions in economic analysis.Present
WorthIn a present worthcomparison of alternatives, thecosts associated with each
alternative investment are allconverted to a present sum of money, and the least ofthese
values represents the best alternative. Annual costs,future payments, and gradients must be
brought to thepresent. Converting all cash flows to present worth is of-ten referred to as
discounting.EXAMPLE:Two alternate plans are available for increasing the ca-pacity of
existing water transmission lines between an un-limited source and a reservoir. The
unlimited source is at ahigher elevation than the reservoir. Plan Acalls for thecon-struction
of a parallel pipeline and for flow by gravity. PlanB specifies construction of a booster
pumping station. Esti-mated cost data for the two plans are as follows:Plan APlan
BPipelinePumping StationConstruction cost$1,000,000$200,000Life40 years40 years
(structure)20 years (equipment)Cost of replacing equipment at the end of 20

7. yr0$75,000Operating costs$1000/yr$50,000/yrIf money is worth 12 percent, which plan is
more eco-nomical? (Assume annual compounding, zero salvagevalue, and all other costs
equal for both plans.)Present worth (Plan A)
CP=A(P/A,12%,40)C$1,000,000=$1000(8.24378)C$1,008,244Present worth (Plan
B)CP=A(P/A,12%,40)=F(P/F,12%,20)C$200,000=$50,000(8.24378)=$75,000(0.10367)C$6
19,964Thus, plan B is the least-cost alternative.Asignificant limitation of present worth
analysis isthat it cannot be used to compare alternatives with un-equal economic lives. That
is, a ten-year plan and atwenty-year plan should not be compared by discountingtheir costs
to a present worth. Abetter method of compar-ison is annual cost.Annual CostTo compare
alternatives by annual cost, all cash flowsare changed to a series of uniform payments.
Current ex-penditures, future costs or receipts, and gradients must beconverted to annual
costs. If a lump-sum cash flow occursat some time other than the beginning or end of the
eco-nomic life, it must be converted in a two-step process: firstmoving it to the present and
then spreading it uniformlyover the life of the project.Alternatives with unequal economic
lives may becompared by assuming replacement in kind at the end ofthe shorter life, thus
maintaining the same level of uni-form payment.InsuranceSystem CostPremiumLifePartial
system$8000$100015 yrFull system$15,000$25020 yrEXAMPLE:Compare the value of a
partial or full sprinkler sys-tem purchased at 10 percent interest.Annual cost (partial
system) CA=P(A/P,10%,15)C$1000=$8000(0.13147)C$2051.75Annual cost (full system)
CA=P(A/P,10%,20)C$250=$15,000(0.11746)C$2011.90The full system is slightly more
economically desirable.When costs are this comparable, it is especially importantto
consider other relevant decision criteria, for example,uninsured losses.Rate of ReturnRate
of return is, by definition, the interest rate atwhich the present worth of the net cash flow is
zero. Com-putationally, this method is the most complex method ofcomparison. If more
than one interest factor is involved,5–98Fire Risk Analysis05-07.QXD 11/14/2001 11:53
AM Page 98
the solution is by trial and error. Microcomputer programsare most useful with this
method.The calculated interest rate may be compared to adiscount rate identified as the
“minimum attractive rateof return” or to the interest rate yielded by alternatives.Rate-of-
return analysis is useful when the selection of anumber of projects is to be undertaken
within a fixed orlimited capital budget.EXAMPLE:An industrial fire fighting truck costs
$100,000. Sav-ings in insurance premiums and uninsured losses fromthe acquisition and
operation of this equipment is esti-mated at $60,000/yr. Salvage value of the apparatus
after5 yr is expected to be $20,000. Afull-time driver duringoperating hours will accrue an
added cost of $10,000/yr.What would the rate of return be on this investment?@ 40%
present
worthCP=F(P/F,40%,5)=A(P/A,40%,5)C>$100,000=$20,000(0.18593)=($60,000>10,000)(
2.0352)C$5,478.60@ 50% present
worthCP=F(P/F,50%,5)=A(P/A,50%,5)C>$100,000=$20,000(0.13169)=($60,000>$10,000)
(1.7366)C>$10,536.40By linear interpolation, the rate of return is 43 percent.Benefit-Cost
AnalysisBenefit-cost analysis, also referred to as cost-benefitanalysis, is a method of
comparison in which the conse-quences of an investment are evaluated in monetaryterms
and divided into the separate categories of benefitsand costs. The amounts are then

8. converted to annualequivalents or present worths for comparison.The important steps of a
benefit-cost analysis are1.Identification of relevant benefits and costs2.Measurement of
these benefits and costs3.Selection of best alternative4.Treatment of
uncertaintyIdentification of Relevant Benefits and CostsThe identification of benefits and
costs depends onthe particular project under consideration. Thus, in thecase of fire
prevention or control activities, the benefitsare based on fire losses prior to such activities.
Fire lossesmay be classified as direct or indirect. Direct economiclosses are property and
contents losses. Indirect losses in-clude such things as the costs of injuries and deaths,
costsincurred by business or industry due to business inter-ruption, losses to the
community from interruption of ser-vices, loss of payroll or taxes, loss of market share,
andloss of reputation. The direct costs of fire protection activ-ities include the costs of
constructing fire-resistive build-ings, installation costs of fire protection systems, and
thecosts of operating fire departments. Indirect costs aremore difficult to measure. They
include items such as theconstraints on choice due to fire protection requirementsby state
and local agencies.Amajor factor in the identification of relevant bene-fits and costs pertains
to the decision unit involved. Thus,if the decision maker is a property owner, the
relevantbenefits from fire protection are likely to be the reductionin fire insurance
premiums and fire damage or businessinterruption losses not covered by insurance. In the
caseof a municipality, relevant benefits are the protection ofmembers of the community,
avoidance of tax and pay-rolllosses, and costs associated with assisting fire victims.Potential
benefits, in these instances, are considerablygreater than those faced by a property owner.
However,the community may ignore some external effects of fireincidents. For example, the
1954 automobile transmissionplant fire in Livonia, Michigan, affected the
automobileindustry in Detroit and various automobile dealersthroughout the United States.
However, there was littleincentive for the community to consider such potentiallosses in
their evaluation of fire strategies, since theywould pertain to persons outside the
community. It mightbe concluded, therefore, that the more comprehensive thedecision unit,
the more likely the inclusion of all relevantcosts and benefits, in particular, social costs and
benefits.Measurement of Benefits and CostsDirect losses are measured or estimated
statisticallyor by a priori judgment. Actuarial fire-loss data collectednationally or for a
particular industry may be used, pro-viding it is adequately specific and the collection
mecha-nism is reliable. More often, an experienced judgment ofpotential losses is made,
sometimes referred to as the max-imum probable loss(MPL).Indirect losses, if considered,
are much more difficultto appraise. Apercentage or multiple of direct losses issometimes
used. However, when indirect loss is an im-portant decision parameter, a great deal of
research intomonetary evaluation may be necessary. Procedures forvaluing a human life
and other indirect losses are dis-cussed in Ramachandran.3In the measurement of benefits,
it is appropriate toadjust for utility or disutility which may be associatedwith a fire
loss.Costs may be divided into two major categories:(1)costs of private fire protection
services, and (2)costs ofpublic fire protection services. In either case, cost esti-mates will
reflect the opportunity cost of providing theservice. For example, the cost of building a fire-
resistivestructure is the production foregone due to the diversionof labor and resources to
make such a structure. Similarly,the cost of a fire department is the loss of other commu-

9. nity services which might have been provided with theresources allocated to the fire
department.Selection of Best AlternativeThere are two considerations in determining
benefit-cost criteria. The first pertains to project acceptability,while the second pertains to
project selection.Project acceptability may be based on benefit-cost dif-ference or benefit-
cost ratio. Benefit-cost ratio is a measureEngineering Economics5–9905-07.QXD
11/14/2001 11:53 AM Page 99
of project worth in which the monetary equivalent bene-fits are divided by the monetary
equivalent costs. The firstcriterion requires that the value of benefits less costs begreater
than zero, while the second criterion requires thatthe benefit-cost ratio be greater than
one.The issue is more complicated in the case of projectselection, since several alternatives
are involved. It is nolonger a question of determining the acceptability of asingle project, but
rather selecting from among alternativeprojects. Consideration should be given to changes
incosts and benefits as various strategies are considered.Project selection decisions are
illustrated in Figure 5-7.3.The degree of fire protection is given on the horizontalaxis, while
the marginal costs and benefits associatedwith various levels of fire safety are given on the
verticalaxis. As the diagram indicates, marginal costs are low ini-tially and then increase.
Less information is available con-cerning the marginal benefit curve, and it may, in fact,
behorizontal. The economically optimum level of fire pro-tection is given by the intersection
of the marginal costand marginal benefit curves. Beyond this point, benefitsfrom increasing
fire protection are exceeded by the costsof providing the additional safety.Anumerical
example is given in Table 5-7.3. There arefive possible strategies or programs possible. The
firststrategy, A, represents the initial situation, while the re-maining four strategies
represent various fire loss reduc-tion activities, each with various costs. Strategies
arearranged in ascending order of costs. Fire losses undereach of the five strategies are
given in the second row,while the sum of fire losses and fire reduction costs foreach
strategy is given in the third row. The sum of firelosses and fire reduction costs of each
strategy is equiva-lent to the life-cycle cost of that strategy. Life-cycle costanalysis is an
alternative to benefit-cost analysis when theoutcomes of the investment decision are cost
savingsrather than benefits per se. Additional information onlife-cycle cost analysis is found
in Fuller and Petersen.5Data in the first two rows may then be used to deter-mine the
marginal costs or marginal benefits from the re-placement of one strategy by another. Thus,
strategy Bhas a fire loss of $70 compared to $100 for strategy A, sothe marginal benefit is
$30. Similarly, the marginal benefitfrom strategy C is the reduction in fire losses from B to
Cor $20. The associated marginal cost of strategy C is $15.Declining marginal benefits and
rising marginal costs re-sult in the selection of strategy C as the optimum strategy.At this
point, the difference between marginal benefitsand marginal costs is still positive.Marginal
benefit-cost ratios are given in the last row.It is worth noting that, while the highest
marginal benefit-cost ratio is reached at activity level B (as is the highestmarginal benefit-
cost difference), project C is still opti-mum, since it yields an additional net benefit of $5.
Thisfinding is reinforced by examining changes in the sum offire losses and fire reduction
costs (i.e., life-cycle costs).Total cost plus loss first declines, reaching a minimum atpoint C,
and then increases. This pattern is not surprising,since as long as marginal benefits exceed
marginal costs,total losses should decrease. Thus, the two criteria—equating marginal costs

10. and benefits, and minimizing thesum of fire losses and fire reduction costs—yield
identicaloutcomes.Treatment of UncertaintyAfinal issue concerns the treatment of
uncertainty.One method for explicitly introducing risk considerationsis to treat benefits and
costs as random variables whichmay be described by probability distributions. For exam-
ple, an estimate of fire losses might consider the followingevents: no fire, minor fire,
intermediate fire, and major fire.Each event has a probability of occurrence and an associ-
ated damage loss. The total expected loss (EL) is given byELC}3iC0piDiwherep0Cprobability
of no firep1Cprobability of a minor fire5–100Fire Risk AnalysisMarginal costsMarginal
benefits0$100%Degree of fire safetyFigure 5-7.3.Project selection.CategoryFire reduction
costsFire lossesSum of fire reductioncosts and fire lossesMarginal benefitsMarginal
costsMarginal benefits minusmarginal costsMarginal benefit-cost
ratioStrategyA0100100000—B1070803010203.0C255075201551.33D4540851020–
100.5E7035105525–200.2Table 5-7.3Use of Benefit-Cost Analyses in StrategySelection05-
07.QXD 11/14/2001 11:53 AM Page 100
p2Cprobability of an intermediate firep3Cprobability of a major fireDnCassociated damage
loss, nC0,1,2,3Expected losses may be computed for different fireprotection strategies.
Thus, a fire protection strategy thatcosts C3and reduces damage losses of a major fire from
D3to D3will result in an expected lossELCp0D0=p1D1=p2D2=p3D3=C3Similarly, a fire
control strategy that costs C2and re-duces the probability of an intermediate fire from p2to
p2has an expected lossELCp0D0=p1D1=p2D2=p3D3=C2Acomparison of expected losses
from alternativestrategies may then be used to determine the optimalstrategy.Use of
expected value has a limitation in that only theaverage value of the probability distribution
is consid-ered. Discussion of other procedures for evaluating un-certain outcomes is given
by Anderson and Settle.6References Cited1.ASTM E833, Definitions of Terms Relating to
Building Econom-ics,American Society for Testing and Materials, West Con-shohocken,
PA(1999).2.ASTM Standards on Building Economics,4th ed., American Soci-ety for Testing
and Materials, West Conshohocken, PA(1999).3.G. Ramachandran, The Economics of Fire
Protection, E & FNSpon, London (1998).4.American National Standards Institute Standard
Z94.0-1982,“Industrial Engineering Terminology,” Chapter 5, Engineer-ing
Economy,Industrial Engineering and Management Press,Atlanta, GA(1983).5.S.K. Fuller and
S.R. Petersen, “Life-Cycle Costing Manualfor the Federal Energy Management Program,”
NIST Hand-book 135,National Institute of Standards and Technology,Gaithersburg, MD
(1996).6.L.G. Anderson and R.E. Settle, Benefit-Cost Analysis: APracti-cal Guide,Lexington
Books, Lexington, MA(1977).Additional ReadingsR.E. Chapman, “ACost-Conscious Guide to
Fire Safety in HealthCare Facilities,” NBSIR 82-2600,National Bureau of Stan-dards,
Washington, DC (1982).R.E. Chapman and W.G. Hall, “Code Compliance at LowerCosts:
AMathematical Programming Approach,” Fire Tech-nology, 18, 1, pp.77–89 (1982).L.P.
Clark, “ALife-Cycle Cost Analysis Methodology for Fire Pro-tection Systems in New Health
Care Facilities,” NBSIR 82-2558,National Bureau of Standards, Washington, DC (1982).W.J.
Fabrycky, G.J. Thuesen, and D. Verma, Economic DecisionAnalysis, 3rd ed., Prentice Hall
International, London (1998).E.L. Grant, W.G. Ireson, and R.S. Leavenworth, Principles of
Engi-neering Economy, 8th ed., John Wiley and Sons, New York(1990).J.S. McConnaughey,
“An Economic Analysis of Building CodeImpacts: ASuggested Approach,” NBSIR 78-

11. 1528,NationalBureau of Standards, Washington, DC (1978).D.G. Newnan and J.P. Lavelle,
Engineering Economic Analysis, 7thed., Engineering Press, Austin, TX (1998).C.S. Park,
Contemporary Engineering Economics, 2nd ed., Addison-Wesley, Menlo Park, CA(1997).J.L.
Riggs, D.D. Bedworth, and S.U. Randhawa, Engineering Eco-nomics, 4th ed., McGraw-Hill,
New York (1996).R.T. Ruegg and H.E. Marshall, Building Economics: Theory andPractice,
Van Nostrand Reinhold, New York (1990).R.T. Ruegg and S.K. Fuller, “ABenefit-Cost Model
of ResidentialFire Sprinkler Systems,” NBS Technical Note 1203,NationalBureau of
Standards, Washington, DC (1984).W.G. Sullivan, J.A. Bontadelli, and E.M. Wicks,
Engineering Econ-omy, 11th ed., Prentice Hall, Upper Saddle River, NJ (2000).Engineering
Economics5–101Appendix A:Symbols and Definitions of Economic ParametersAjCash flow
at end of period jAEnd-of-period cash flows (or equivalent end-of-period values) in a
uniform series continuing for a specified number of periodsA–Amount of money (or
equivalent value) flowing continuously and uniformly during each period, continuing for a
specified numberof periodsFFuture sum of money—the letter Fimplies future (or
equivalent future value)GUniform period-by-period increase or decrease in cash flows (or
equivalent values);the arithmetic gradientMNumber of compounding periods per interest
periodaNNumber of compounding periodsPPresent sum of money—the letter “P”implies
present (or equivalent present value).Sometimes used to indicate initial
capitalinvestment.SSalvage (residual) value of capital investmentfRate of price level
increase or decrease per period;an “inflation”of “escalation”rategUniform rate of cash flow
increase or decrease from period to period;the geometric gradientiEffective interest rate
per interest perioda(discount rate), expressed as a percent or decimal fractionrNominal
interest rate per interest period,aexpressed as a percent or decimal fractionaNormally, but
not always, the interest period is taken as 1 yr.Subperiods, then, would be quarters, months,
weeks, and so forth.SymbolDefinition of Parameter05-07.QXD 11/14/2001 11:53 AM Page
101
5–102Fire Risk AnalysisName of FactorGroup A.All cash flows discrete:end-of-period
compoundingCompound amount (single payment)Present worth (single payment)Sinking
fundCapital recoveryCompound amount (uniform series)Present worth (uniform
series)Arithmetic gradient to uniform seriesArithmetic gradient to present worthGroup
B.All cash flows discrete:continuous compounding at nominal rate r per periodContinuous
compounding compound amount (single payment)Continuous compounding present worth
(single payment)Continuous compounding present worth (single payment)Continuous
compounding sinking fundContinuous compounding capital recoveryContinuous
compounding compound amount (uniform series)Group C.Continuous, uniform cash
flows:continuouscompoundingContinuous compounding sinking fund (continuous,uniform
payments)Continuous compounding capital recovery (continuous,uniform
payments)Continuous compounding compound amount (continuous,uniform
payments)Continuous compounding present worth (continuous,uniform
payments)AlgebraicFormulation(1 =i)N(1 =i)–Ni(1 =i)N– 1i(1 =i)N(1 =i)N– 1(1 =i)N– 1i(1
=i)N– 1i(1 =i)N(1 =i)N– iN– 1i(1 =i)N– i(1 =i)N– iN– 1i2(1 =i)NerNe–rNerN– 1erN(er– 1)er–
1erN– 1erN(er– 1)erN– 1erN– 1er– 1rerN– 1rerNerN– 1erN– 1rerN–
1rerNFunctionalFormat(F/P, i, N)(P/F, i, N)(A/F, i, N)(A/P, i, N)(F/A, i, N)(P/A, i, N)(A/G, i,

12. N)(P/G, i, N)(F/P, r, N)(P/F, r, N)(P/A, r, N)(A/F, r, N)(A/P, r, N)(F/A, r, N)A–/F, r, NA–/P,
r,NF/A–, r, N)P/A–, r, N)Appendix B:Functional Forms of Compound Interest FactorsaaSee
Appendix A for definitions of symbols used in this table.05-07.QXD 11/14/2001 11:53 AM
Page 102
Appendix C—Interest TablesEngineering Economics5–
103Yr.1234567891011121314151617181920212223242530354045506070809010030%
.7692.5917.4552.3501.2693.2072.1594.1226.0943.0725.0558.0429.0330.0254.0195.0150.
0116.0089.0068.0053.0040.0031.0024.0018.001425%.8000.6400.5120.4096.3277.2621.2
097.1678.1342.1074.0859.0687.0550.0440.0352.0281.0225.0180.0144.0115.0092.0074.00
59.0047.0038.0012.0004.000120%.8333.6944.5787.4823.4019.3349.2791.2326.1938.161
5.1346.1122.0935.0779.0649.0541.0451.0376.0313.0261.0217.0181.0151.0126.0105.0042
.0017.0007.0003.000115%.8696.7561.6575.5718.4972.4323.3759.3269.2843.2472.2149.1
869.1625.1413.1229.1069.0929.0808.0703.0611.0531.0462.0402.0349.0304.0151.0075.00
37.0019.0009.000212%.8929.7972.7118.6355.5674.5066.4523.4039.3606.3220.2875.256
7.2292.2046.1827.1631.1456.1300.1161.1037.0926.0826.0738.0659.0588.0334.0189.0107
.0061.0035.0011.000410%.9091.8264.7513.6830.6209.5645.5132.4665.4241.3855.3505.3
186.2897.2633.2394.2176.1978.1799.1635.1486.1351.1228.1117.1015.0923.0573.0356.02
21.0137.0085.0033.0013.0005.00028%.9259.8573.7938.7350.6806.6302.5835.5403.5002.
4632.4289.3971.3677.3405.3152.2919.2703.2502.2317.2145.1987.1839.1703.1577.1460.0
994.0676.0460.0313.0213.0099.0046.0021.0010.00056%.9434.8900.8396.7921.7473.705
0.6651.6274.5919.5584.5268.4970.4688.4423.4173.3936.3714.3503.3305.3118.2942.2775
.2618.2470.2330.1741.1301.0972.0727.0543.0303.0169.0095.0053.00304%.9615.9246.88
90.8548.8219.7903.7599.7307.7026.6756.6496.6246.6006.5775.5553.5339.5134.4936.474
6.4564.4388.4220.4057.3901.3751.3083.2534.2083.1712.1407.0951.0642.0434.0293.0198
2%.9804.9612.9423.9238.9057.8880.8706.8535.8368.8203.8043.7885.7730.7579.7430.72
84.7142.7002.6864.6730.6698.6468.6342.6217.6095.5521.5000.4529.4102.3715.3048.250
0.2051.1683.138040%.7143.5102.3644.2603.1859.1328.0949.0678.0484.0346.0247.0176.
0126.0090.0064.0046.0033.0023.0017.0012.0009.0006.0004.0003.000250%.6667.4444.2
963.1975.1317.0878.0585.0390.0260.0173.0116.0077.0051.0034.0023.0015.0010.0007.00
05.0003Table C-7.1Present Worth Factor (Changes F to P)1(1 =i)y05-07.QXD 11/14/2001
11:53 AM Page 103
5–104Fire Risk
AnalysisYr.12345678910111213141516171819202122232425303540455060708090100
30%1.300.7348.5506.4616.4106.3784.3569.3419.3312.3235.3177.3135.3102.3078.3060.3
046.3035.3027.3021.3016.3012.3009.3007.3006.3004.300125%1.250.6944.5123.4234.37
19.3388.3163.3004.2888.2801.2735.2685.2645.2615.2591.2572.2558.2546.2537.2529.252
3.2519.2515.2512.2510.2503.2501.250020%1.200.6545.4747.3863.3344.3007.2774.2606.
2481.2385.2311.2253.2206.2169.2139.2114.2094.2078.2065.2054.2044.2037.2031.2025.2
021.2008.2003.2001.2001.200015%1.150.6151.4380.3503.2983.2642.2404.2229.2096.19
93.1911.1845.1791.1747.1710.1679.1654.1632.1613.1598.1584.1573.1563.1554.1547.152
3.1511.1506.1503.1501.150012%1.120.5917.4163.3292.2774.2432.2191.2013.1877.1770.
1684.1614.1557.1509.1469.1434.1405.1379.1358.1339.1322.1308.1296.1285.1275.1241.1
223.1213.1207.1204.120010%1.100.5762.4021.3155.2638.2296.2054.1874.1736.1627.15

13. 40.1468.1408.1357.1315.1278.1247.1219.1195.1175.1156.1140.1126.1113.1102.1061.103
7.1023.1014.1009.1003.1001.10008%1.080.5608.3880.3019.2505.2163.1921.1740.1601.1
490.1401.1327.1265.1213.1168.1130.1096.1067.1041.1019.0998.0980.0964.0950.0937.08
88.0858.0839.0826.0817.0808.0804.0802.0801.08006%1.060.5454.3741.2886.2374.2034.
1791.1610.1470.1359.1268.1193.1130.1076.1030.0990.0954.0924.0896.0872.0850.0830.0
813.0797.0782.0726.0690.0664.0647.0634.0619.0610.0606.0603.06024%1.040.5302.360
3.2755.2246.1908.1666.1485.1345.1233.1141.1066.1001.0947.0899.0858.0822.0790.0761
.0736.0713.0692.0673.0656.0640.0578.0536.0505.0483.0466.0442.0428.0418.0412.04082
%1.020.5150.3468.2626.2122.1785.1545.1365.1225.1113.1022.0946.0881.0826.0778.073
7.0700.0667.0638.0611.0588.0566.0547.0529.0512.0446.0400.0366.0339.0318.0288.0267
.0252.0241.023240%1.400.8167.6294.5408.4914.4613.4419.4291.4203.4143.4101.4072.4
051.4036.4026.4019.4013.4009.4007.4005.4003.4002.4002.4001.4001.400050%1.500.90
00.7105.6231.5758.5481.5311.5203.5134.5088.5059.5039.5026.5017.5011.5008.5005.500
3.5002.5002.5000Table C-7.2Capital Recovery Factor (Changes P to A)i(1 =i)y(1 =i)y– 105-
07.QXD 11/14/2001 11:53 AM Page 104