Description

View attached document. Response to each post.THE TOPIC: Why would engineers be involved in cost analysis?? Isn’t this better left to

accountants or CEO type people??

POST1:

Engineers are involved in cost analysis because there are many things in engineering that

require knowing specifics. Accountants may look at cost as one simple blanket cost per sq ft.

As I have understood, many general contractors will do this to get a rough ballpark type price.

While this can work out okay in typical simple projects, it will likely not provide you with a very

accurate number for other projects. An engineer would be able to get a more detailed bid with

labor, materials such as pipe lengths, type of pipe, fittings, specialty sprinklers, fire pumps etc.

A building with a dry system will not cost the same as if it was a wet system. More intricate

ceilings and building designs can increase the number of sprinklers significantly on a project.

Specialty sprinklers like window sprinklers or attic sprinklers that are a lot more costly may be

needed..

RESPONSE EXAMPLE:

Exactly my thoughts Jessica. Accountants might have a good knowlegde of costs calculations,

but not how different dimensions of a construction project impact the costs as well as quality. Besides

maintaining the costs, engineers monitor and safeguard the equipment used on a construction project.

They are responsible for the integrity and safety of this equipment. Thus, they can provide a better picture

of costs associated with a project.

.

YOU RESPONSE TO POST1:

2-3 SENTENCES

POST2:

Cost analysis helps determine a project’s feasibility and the anticipated profits. The

managerial and accounting departments do a general overview of the involved costs of

production but lack a deeper understanding of material type, labor, and workforce for

successful completion. When analyzing the price of a project, engineers would be involved

because they will give a more comprehensive look into the operating costs and find the bestsuited financial option within the stipulated budget. Engineers also have a more accurate

estimate of when a project is due given the suitable materials and human resources; this saves

cost and prevents last-minute risks from financial strain (Project Management; Cost Estimation,

2023). Also, the company executives are more focused on increasing company income hence

involving the engineer in cost breakdown will improve efficiency in delivering the top benefits

of a project on time, attracting more clientele.

References

Project Management for Construction: Cost Estimation. (2023). Cmu.edu.

https://www.cmu.edu/cee/projects/PMbook/05_Cost_Estimation.html

RESPONSE EXAMPLE:

Great post! One thing I would like to add is calculated losses. There are many projects out there

that end up losing money because of no proper planning and some mistakes along the way! I think a

good engineer can utilize his/her experience to validate why one should use a certain manufacturer over

another by past experience. I think a lot of companies would like to profit but some would just like to break

even because too many issues during the process. Well done!

YOUR RESPONSE TO POST2:

2-3 SENTENCES

THE TOPIC: You are going to buy a new car (an awesome one – I might add). Review some

websites and go over their various options (buy, lease, 0% APR vs. cash back, etc.). When would

these options make a difference? When would they be more beneficial? Why?

POST 2.1:

The economy in whichever way you look at it has its own pros anfd cons. Therefore I

would say that whether you lease a car, buy a car, have a 0% APR, or get cash back on your

purchases, all depend on your situation.

When you lease a car, some of the issues you deal with are having to deal with the sale price, a

certain lenght of the period of lease, maximum amaount of mileage allowed, no ownership,

fees and other costs, lack of control, but you also get away with low monthly costs, a new car

every few years, worry-free maintenance, no resale worries, and you may even enjoy some tax

deductions.

When you buy and own a car, it is the other way round leasing but in this case you will also

have to deal with rapid depreciation that will affect resale, and the immense amount of driving

and maintenance costs.

So, one cap does not fit all when it comes to economical benefit because every situation is

different.

RESPONSE EXAMPLE:

HiI 100% agree with you! I think a lot of it also depends on the economy too! I think, at the

moment, cash is king and if someone is capable of getting a loan for less than 5%, they are in the green

because of current inflation and interest rates. I have always been a cash buyer, but I definitely rethought

that in our current economy! Great post!

YOUR RESPONSE TO POST 2.1:

2-3 SENTENCES.

POST 2.2:

Today, cars and other automobiles are one of the most common necessities. The

discussion of purchasing a new car can be quite unsettling due to current prices as well as

increased demand. Understanding your options, the market, and other factors will in theory

make your next purchase easier. Most dealerships allow three forms of payment those being

leasing, financing, and purchasing. Leasing a car, like a long-term rental, requires an upfront

and monthly payment system. At the end of the lease, you have the option to decide to

continue that lease, purchase, or even change cars. Financing your car involves taking out a

loan to purchase the vehicle and then paying back that loan over time. Applied interest rates

and fees are involved. Finally, purchasing your vehicle simply means paying full price for the

vehicle on the lot. At the end of the day, all options have their unique pros and cons but the

most important factor to consider is what can you and your budget afford and whether is it

within your parameters.

RESPONSE EXAMPLE:

You make a good point here! Because cars are more of a necessity than luxury to many,

the control of which acquisition joyce is limited even though the determining factors are many.

In the investment perspective, the markets really determine which way is more beneficial than

the other. When interest rates are low borrowing is not a big problem as compared to when

interest rates are high. And again individual situations would determine the best beneficial

options for them.

Thanks for your post.

YOUR RESPONSE TO POST 2.2:

2-3 SENTENCES.

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Page 93

SECTION FIVE

CHAPTER 7

Engineering Economics

John M. Watts, Jr., and Robert E. Chapman

Introduction

Engineering economics is the application of economic

techniques to the evaluation of design and engineering alternatives.1 The role of engineering economics is to assess

the appropriateness of a given project, estimate its value,

and justify it from an engineering standpoint.

This chapter discusses the time value of money and

other cash-flow concepts, such as compound and continuous interest. It continues with economic practices and

techniques used to evaluate and optimize decisions on selection of fire safety strategies. The final section expands

on the principles of benefit-cost analysis.

An in-depth treatment of the practices and techniques

covered in this compilation is available in the ASTM

compilation of standards on building economics.2 The

ASTM compilation also includes case illustrations showing how to apply the practices and techniques to investment decisions.

A broader perspective on the application of engineering economics to fire protection engineering can be

found in The Economics of Fire Protection by Ramachandran.3 This work is intended as a textbook for fire protection engineers and includes material and references that

expand on several other chapters of this section of the

SFPE handbook.

Cash-Flow Concepts

Cash flow is the stream of monetary (dollar) values—

costs (inputs) and benefits (outputs)—resulting from a

project investment.

Dr. John M. Watts, Jr., holds degrees in fire protection engineering,

industrial engineering, and operations research. He is director of the

Fire Safety Institute, a not-for-profit information, research, and educational corporation located in Middlebury, Vermont. Dr. Watts also

serves as editor of NFPA’s Fire Technology.

Dr. Robert E. Chapman is an economist in the Office of Applied Economics, Building and Fire Research Laboratory, National Institute of

Standards and Technology.

Time Value of Money

The following are reasons why $1000 today is

“worth” more than $1000 one year from today:

1. Inflation

2. Risk

3. Cost of money

Of these, the cost of money is the most predictable,

and, hence, it is the essential component of economic

analysis. Cost of money is represented by (1) money paid

for the use of borrowed money, or (2) return on investment. Cost of money is determined by an interest rate.

Time value of money is defined as the time-dependent

value of money stemming both from changes in the purchasing power of money (inflation or deflation) and from

the real earning potential of alternative investments over

time.

Cash-Flow Diagrams

It is difficult to solve a problem if you cannot see it. The

easiest way to approach problems in economic analysis is

to draw a picture. The picture should show three things:

1. A time interval divided into an appropriate number of

equal periods

2. All cash outflows (deposits, expenditures, etc.) in each

period

3. All cash inflows (withdrawals, income, etc.) for each

period

Unless otherwise indicated, all such cash flows are considered to occur at the end of their respective periods.

Figure 5-7.1 is a cash-flow diagram showing an outflow or disbursement of $1000 at the beginning of year 1

and an inflow or return of $2000 at the end of year 5.

Notation

To simplify the subject of economic analysis, symbols are introduced to represent types of cash flows and

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Fire Risk Analysis

$2000

F(2) C F(1) = F(1)(i)

Interest is applied to the new sum:

C (F)(1)(1 = i) C P(1 = i)2

F (3) C F (2)(1 = i) C P(1 = i)3

0

2

1

3

4

5

and by mathematical induction,

F(N) C P(1 = i)N

$1000

Figure 5-7.1.

Cash-flow diagram.

interest factors. The symbols used in this chapter conform

to ANSI Z94;4 however, not all practitioners follow this

standard convention, and care must be taken to avoid

confusion when reading the literature. The following

symbols will be used here:

P C Present sum of money ($)

F C Future sum of money ($)

N C Number of interest periods

i C Interest rate per period (%)

A complete list of the ANSI Z94 symbols is given in Appendix A to this chapter.

Interest Calculations

Interest is the money paid for the use of borrowed

money or the return on invested capital. The economic

cost of construction, installation, ownership, or operation

can be estimated correctly only by including a factor for

the economic cost of money.

Simple interest: To illustrate the basic concepts of interest, an additional notation will be used:

F(N) C Future sum of money after N periods

Then, for simple interest,

F(1) C P = (P)(i) C P(1 = i)

EXAMPLE:

$100 at 10 percent per year for 5 yr yields

F(5) C 100(1 = 0.1)5

C 100(1.1)5

C 100(1.61051)

C $161.05

which is over 7 percent greater than with simple interest.

EXAMPLE:

In 1626 Willem Verhulst bought Manhattan Island

from the Canarsie Indians for 60 florins ($24) worth of

merchandise (a price of about 0.5 cents per hectare [0.2

cents per acre]). At an average interest rate of 6 percent,

what is the present value (2001) of the Canarsies’ $24?

F C P(1 = i)N

C $24(1 = 0.06)375

C $7.4 ? 1010

Seventy-four billion dollars is a reasonable approximation of the present land value of the island of Manhattan.

Interest Factors

Interest factors are multiplicative numbers calculated

from interest formulas for given interest rates and periods. They are used to convert cash flows occurring at different times to a common time. The functional formats

used to represent these factors are taken from ANSI Z94,

and they are summarized in Appendix B to this chapter.

and

F(N) C P = (N)(P)(i) C P(1 = Ni)

For example: $100 at 10 percent per year for 5 yr yields

F(5) C 100[1 = (5)(0.1)]

C 100(1.5)

C $150

However, interest is almost universally compounded to

include interest on the interest.

Compound interest

F(1) C P = (P)(i) C P(1 = i)

is the same as simple interest,

Compound Amount Factor

In the formula for finding the future value of a sum of

money with compound interest, the mathematical expression (1 = i)N is referred to as the compound amount factor,

represented by the functional format (F/P, i, N). Thus,

F C P(F/P, i, N)

Interest tables: Values of the compound amount, present worth, and other factors that will be discussed

shortly, are tabulated for a variety of interest rates and

number of periods in most texts on engineering economy.

Example tables are presented in Appendix C to this chapter. Although calculators and computers have greatly reduced the need for such tables, they are often still useful

for interpolations.

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Engineering Economics

Present Worth

Present worth is the value found by discounting future cash flows to the present or base time.

Discounting: The inverse of compounding is determining a present amount which will yield a specified future

sum. This process is referred to as discounting. The equation for discounting is found readily by using the compounding equation to solve for P in terms of F:

Nominal versus effective interest: It is generally assumed that interest is compounded annually. However,

interest may be compounded more frequently. When this

occurs, there is a nominal interest or annual percentage rate

and an effective interest, which is the figure used in calculations. For example, a savings bank may offer 5 percent

interest compounded quarterly, which is not the same as

5 percent per year. A nominal rate of 5 percent compounded quarterly is the same as 1.25 percent every three

months or an effective rate of 5.1 percent per year. If

P C F(1 = i)>N

EXAMPLE:

What present sum will yield $1000 in 5 yr at 10 percent?

P C 1000(1.1)>5

C 1000(0.62092)

C $620.92

This result means that $620.92 “deposited” today at 10

percent compounded annually will yield $1000 in 5 yr.

Present worth factor: In the discounting equation, the

expression (1 = i)>N is called the present worth factor and is

represented by the symbol (P/F, i, N). Thus, for the present

worth of a future sum at i percent interest for N periods,

r C Nominal interest rate,

and

M C Number of subperiods per year

then the effective interest rate is

‹

M

r

>1

iC 1=

M

EXAMPLE:

Credit cards usually charge interest at a rate of 1.5

percent per month. This amount is a nominal rate of 18

percent. What is the effective rate?

i C (1 = 0.015)12 > 1

C 1.1956 > 1

C 19.56%

P C F(P/F, i, N)

Note that the present worth factor is the reciprocal of the

compound amount factor. Also note that

(P/F, i, N) C

1

(F/P, i, N)

EXAMPLE:

What interest rate is required to triple $1000 in 10 years?

PC

F

C (P/F, i, 10)

3

therefore,

(P/F, i, 10) C

1

3

From Appendix C,

(P/F, 10%, 10) C 0.3855

and

Continuous interest: A special case of effective interest

occurs when the number of periods per year is infinite.

This represents a situation of continuous interest, also referred to as continuous compounding. Formulas for continuous interest can be derived by examining limits as M

approaches infinity. Formulas for interest factors using

continuous compounding are included in Appendix B.

Continuous interest is compared to monthly interest in

Table 5-7.1.

EXAMPLE:

Compare the future amounts obtained under various

compounding periods at a nominal interest rate of 12 percent for 5 yr, if P C $10,000. (See Table 5-7.2.)

Series Payments

Life would be simpler if all financial transactions were

in single lump-sum payments, now or at some time in the

(P/F, 12%, 10) C 0.3220

Table 5-7.1

By linear interpolation,

Continuous Interest (%)

Effective

i C 11.6%

Interest Periods

Normally, but not always, the interest period is taken

as 1 yr. There may be subperiods of quarters, months,

weeks, and so forth.

Nominal %

Monthly

Continuous

5

10

15

20

5.1

10.5

16.1

21.9

5.1

10.5

16.2

22.1

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Fire Risk Analysis

Table 5-7.2

Example of Continuous Interest N C 5 yr, r C 12%

Compounding

M

i

NM

F/P

F

Annual

Semi-annual

Quarterly

Monthly

Weekly

Daily

Hourly

Instantaneously

1

2

4

12

52

365

8760

ã

12.000

12.360

12.551

12.683

12.734

12.747

12.749

12.750

5

10

20

60

260

1825

43,800

ã

1.76234

1.79085

1.80611

1.81670

1.820860

1.821938

1.822061

1.822119a

17,623.40

17,908.50

18,061.10

18,167.00

18,208.60

18,219.38

18,220.61

18,221.19

aF/P (instantaneous) C e Ni C e 5(0.12) C e 0.6.

Series compound amount factor: Given a series of regular payments, what will they be worth at some future time?

Let

Capital recovery factor: It is also important to be able to

relate 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 C P(F/P, i, N)

in the sinking fund equation, A C F(A/F, i, N), yields

A C the amount of a regular end-of-period payment

A C P(F/P, i, N)(A/F, i, N)

Then, note that each payment, A, is compounded for a

different period of time. The first payment will be compounded for N > 1 periods (yr):

And, substituting the corresponding interest factors gives

future. However, most situations involve a series of regular payments, for example, car loans and mortgages.

F C A(1 = i)N>1

and the second payment for N > 2 periods:

F C A(1 = i)N>2

AC P

In this equation, the interest expression is known as the

capital recovery factor, since the equation defines a regular

income necessary to recover a capital investment. The

symbolic equation is

A C P(A/P, i, N)

and so forth. Thus, the total future value is

F C A(1 = i)N>1 = A(1 = i)N>2 = ß = A(1 = i) = A

or

FC

A[(1 = i)N > 1]

i

The interest expression in this equation is known as the

series compound amount factor, (F/A, i, N), thus

F C A(F/A, i, N)

Sinking fund factor: The process corresponding to the

inverse of series compounding is referred to as a sinking

fund; that is, what size regular series payments are necessary to acquire a given future amount?

Solving the series compound amount equation for A,

8

4

i

AC F

[(1 = i)N > 1]

Or, using the symbol (A/F, i, N) for the sinking fund factor

A C F(A/F, i, N)

Here, note that the sinking fund factor is the reciprocal of

the series compound amount factor, that is, (A/F, i, N) C

1/(F/A, i, N).

[i(1 = i)N ]

[(1 = i)N > 1]

Series present worth factor: As with the other factors,

there is a corresponding inverse to the capital recovery

factor. The series present worth factor is found by solving

the capital recovery equation for P.

PCA

[(1 = i)N > 1]

[i(1 = i)N ]

or, symbolically

P C A(P/A, i, N)

Other Interest Calculation Concepts

Additional concepts involved in interest calculations

include continuous cash flow, capitalized costs, beginning of period payments, and gradients.

Continuous cash flow: Perhaps the most useful function of continuous interest is its application to situations

where the flow of money is of a continuous nature. Continuous cash flow is representative for

1. A series of regular payments for which the interval between payments is very short

2. A disbursement at some unknown time (which is then

considered to be spread out over the economic period)

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Engineering Economics

Factors for calculating present or future worth of a

series of annual amounts, representing the total of a continuous cash flow throughout the year, may be derived

by integrating corresponding continuous interest factors

over the number of years the flow is maintained.

Continuous cash flow is an appropriate way to handle economic evaluations of risk, for example, the present

value of an annual expected loss.

Formulas for interest factors representing continuous, uniform cash flows are included in Appendix B.

Fire protection engineering economic analysis is primarily concerned with cost-reduction decisions, finding

the least expensive way to fulfill certain requirements, or

minimizing the sum of expected fire losses plus investment in fire protection.

There are four common methods of comparing alternative investments: (1) present worth, (2) annual cost,

(3) rate of return, and (4) benefit-cost analysis. Each of

these is dependent on a selected interest rate or discount

rate to adjust cash flows at different points in time.

Capitalized costs: Sometimes there are considerations,

such as some public works projects, which are considered

to last indefinitely and thereby provide perpetual service.

For example, how much should a community be willing

to invest in a reservoir which will reduce fire insurance

costs by some annual amount, A? Taking the limit of the

series present worth factor as the number of periods goes

to infinity gives the reciprocal of the interest rate. Thus,

capitalized costs are just the annual amount divided by the

interest rate. When expressed as an amount required to

produce a fixed yield in perpetuity, it is sometimes referred to as an annuity.

Discount Rate

Beginning-of-period payments: Most returns on investment (cash inflows) occur at the end of the period during

which they accrued. For example, a bank computes and

pays interest at the end of the interest period. Accordingly,

interest tables, such as those in Appendix C, are computed

for end-of-year payments. For example, the values of the

capital recovery factor (A/P, i, N) assume that the regular

payments, A, occur at the end of each period.

On the other hand, most disbursements (cash outflows) occur at the beginning of the period (e.g., insurance

premiums). When dealing with beginning-of-period payments, it is necessary to make adjustments. One method

of adjustment for beginning-of-period payments is to calculate a separate set of factors. Another way is to logically

interpret the effect of beginning-of-period payments for a

particular problem, for example, treating the first payment as a present value. The important thing is to recognize that such variations can affect economic analysis.

Gradients: It occasionally becomes necessary to treat

the case of a cash flow which regularly increases or decreases at each period. Such patterned changes in cash

flow are called gradients. They may be a constant amount

(linear or arithmetic progression), or they may be a constant percentage (exponential or geometric progression).

Various equations for dealing with gradient series may be

found in Appendix B.

The term discount rate is often used for the interest

rate when comparing alternative projects or strategies.

Selection of discount rate: If costs and benefits accrue

equally over the life of a project or strategy, the selection

of discount rate will have little impact on the estimated

benefit-cost ratios. However, most benefits and costs occur at different times over the project life cycle. Thus,

costs of constructing a fire-resistive building will be incurred early in contrast to benefits, which will accrue over

the life of the building. The discount rate then has a significant impact on measures such as benefit-cost ratios,

since the higher the discount rate, the lower the present

value of future benefits.

In view of the uncertainty concerning appropriate discount rate, analysts frequently use a range of discount

rates. This procedure indicates the sensitivity of the analysis to variations in the discount rate. In some instances,

project rankings based on present values may be affected

by the discount rate as shown in Figure 5-7.2. Project A is

preferred to project B for discount rates below 15 percent,

while the converse is true for discount rates greater than

15 percent. In this instance, the decision to adopt project A

in preference to project B will reflect the belief that the appropriate discount rate is less than or equal to 15 percent.

A

Net present value

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B

Comparison of Alternatives

Most decisions are based on economic criteria. Investments are unattractive, unless it seems likely they will

be recovered with interest. Economic decisions can be divided into two classes:

1. Income-expansion—that is, the objective of capitalism

2. Cost-reduction—the basis of profitability

0

5

10

15

20

Discount rate (%)

Figure 5-7.2. Impact of discount rate on project selection.

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A comparison of benefits and costs may also be used

to determine the payback period for a particular project or

strategy. However, it is important to discount future costs

or benefits in such analyses. For example, an analysis of

the Beverly Hills Supper Club fire compared annual savings from a reduction in insurance premiums to the costs

of 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 installation

would have been paid back in four to seven years (depending on the cost of the sprinklers). However, this

analysis did not discount future benefits, so that $11,000

received at the end of four years was deemed equivalent

to $11,000 received in the first year. Once future benefits

are discounted, the payback period ranges from five to

eleven years with a discount rate of 10 percent.

Inflation and the discount rate: Provision for inflation

may be made in two ways: (1) estimate all future costs

and benefits in constant prices, and use a discount rate

which represents the opportunity cost of capital in the absence of inflation; or (2) estimate all future benefits and

costs in current or inflated prices, and use a discount rate

which includes an allowance for inflation. The discount

rate in the first instance may be considered the real discount rate, while the discount rate in the second instance

is the nominal discount rate. The use of current or inflated

prices with the real discount rate, or constant prices with

the nominal discount rate, will result in serious distortions in economic analysis.

Present Worth

In a present worth comparison of alternatives, the

costs associated with each alternative investment are all

converted to a present sum of money, and the least of

these values represents the best alternative. Annual costs,

future payments, and gradients must be brought to the

present. Converting all cash flows to present worth is often referred to as discounting.

EXAMPLE:

Two alternate plans are available for increasing the capacity of existing water transmission lines between an unlimited source and a reservoir. The unlimited source is at a

higher elevation than the reservoir. Plan A calls for the construction of a parallel pipeline and for flow by gravity. Plan

B specifies construction of a booster pumping station. Estimated cost data for the two plans are as follows:

Plan A

Pipeline

Plan B

Pumping Station

Construction cost

Life

$1,000,000

40 years

$200,000

40 years (structure)

20 years (equipment)

Cost of replacing

equipment at the

end of 20 yr

Operating costs

0

$1000/yr

$75,000

$50,000/yr

If money is worth 12 percent, which plan is more economical? (Assume annual compounding, zero salvage

value, and all other costs equal for both plans.)

Present worth (Plan A) C P = A(P/A, 12%, 40)

C $1,000,000 = $1000(8.24378)

C $1,008,244

Present worth (Plan B) C P = A(P/A, 12%, 40)

= F(P/F, 12%, 20)

C $200,000 = $50,000(8.24378)

= $75,000(0.10367)

C $619,964

Thus, plan B is the least-cost alternative.

A significant limitation of present worth analysis is

that it cannot be used to compare alternatives with unequal economic lives. That is, a ten-year plan and a

twenty-year plan should not be compared by discounting

their costs to a present worth. A better method of comparison is annual cost.

Annual Cost

To compare alternatives by annual cost, all cash flows

are changed to a series of uniform payments. Current expenditures, future costs or receipts, and gradients must be

converted to annual costs. If a lump-sum cash flow occurs

at some time other than the beginning or end of the economic life, it must be converted in a two-step process: first

moving it to the present and then spreading it uniformly

over the life of the project.

Alternatives with unequal economic lives may be

compared by assuming replacement in kind at the end of

the shorter life, thus maintaining the same level of uniform payment.

Partial system

Full system

System Cost

Insurance

Premium

Life

$ 8000

$15,000

$1000

$250

15 yr

20 yr

EXAMPLE:

Compare the value of a partial or full sprinkler system purchased at 10 percent interest.

Annual cost (partial system) C A = P(A/P, 10%, 15)

C $1000 = $8000(0.13147)

C $2051.75

Annual cost (full system) C A = P(A/P, 10%, 20)

C $250 = $15,000(0.11746)

C $2011.90

The full system is slightly more economically desirable.

When costs are this comparable, it is especially important

to consider other relevant decision criteria, for example,

uninsured losses.

Rate of Return

Rate of return is, by definition, the interest rate at

which the present worth of the net cash flow is zero. Computationally, this method is the most complex method of

comparison. If more than one interest factor is involved,

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Engineering Economics

the solution is by trial and error. Microcomputer programs

are most useful with this method.

The calculated interest rate may be compared to a

discount rate identified as the “minimum attractive rate

of return” or to the interest rate yielded by alternatives.

Rate-of-return analysis is useful when the selection of a

number of projects is to be undertaken within a fixed or

limited capital budget.

EXAMPLE:

An industrial fire fighting truck costs $100,000. Savings in insurance premiums and uninsured losses from

the acquisition and operation of this equipment is estimated at $60,000/yr. Salvage value of the apparatus after

5 yr is expected to be $20,000. A full-time driver during

operating hours will accrue an added cost of $10,000/yr.

What would the rate of return be on this investment?

@ 40% present worth

C P = 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 worth

C P = F(P/F, 50%, 5) = A(P/A, 50%, 5)

C >$100,000 = $20,000(0.13169)

= ($60,000 > $10,000)(1.7366)

C >$10,536.40

By linear interpolation, the rate of return is 43 percent.

Benefit-Cost Analysis

Benefit-cost analysis, also referred to as cost-benefit

analysis, is a method of comparison in which the consequences of an investment are evaluated in monetary

terms and divided into the separate categories of benefits

and costs. The amounts are then converted to annual

equivalents or present worths for comparison.

The important steps of a benefit-cost analysis are

1. Identification of relevant benefits and costs

2. Measurement of these benefits and costs

3. Selection of best alternative

4. Treatment of uncertainty

Identification of Relevant Benefits and Costs

The identification of benefits and costs depends on

the particular project under consideration. Thus, in the

case of fire prevention or control activities, the benefits

are based on fire losses prior to such activities. Fire losses

may be classified as direct or indirect. Direct economic

losses are property and contents losses. Indirect losses include such things as the costs of injuries and deaths, costs

incurred by business or industry due to business interruption, losses to the community from interruption of services, loss of payroll or taxes, loss of market share, and

loss of reputation. The direct costs of fire protection activities include the costs of constructing fire-resistive buildings, installation costs of fire protection systems, and the

5–99

costs of operating fire departments. Indirect costs are

more difficult to measure. They include items such as the

constraints on choice due to fire protection requirements

by state and local agencies.

A major factor in the identification of relevant benefits and costs pertains to the decision unit involved. Thus,

if the decision maker is a property owner, the relevant

benefits from fire protection are likely to be the reduction

in fire insurance premiums and fire damage or business

interruption losses not covered by insurance. In the case

of a municipality, relevant benefits are the protection of

members of the community, avoidance of tax and payroll losses, and costs associated with assisting fire victims.

Potential benefits, in these instances, are considerably

greater than those faced by a property owner. However,

the community may ignore some external effects of fire

incidents. For example, the 1954 automobile transmission

plant fire in Livonia, Michigan, affected the automobile

industry in Detroit and various automobile dealers

throughout the United States. However, there was little

incentive for the community to consider such potential

losses in their evaluation of fire strategies, since they

would pertain to persons outside the community. It might

be concluded, therefore, that the more comprehensive the

decision unit, the more likely the inclusion of all relevant

costs and benefits, in particular, social costs and benefits.

Measurement of Benefits and Costs

Direct losses are measured or estimated statistically

or by a priori judgment. Actuarial fire-loss data collected

nationally or for a particular industry may be used, providing it is adequately specific and the collection mechanism is reliable. More often, an experienced judgment of

potential losses is made, sometimes referred to as the maximum probable loss (MPL).

Indirect losses, if considered, are much more difficult

to appraise. A percentage or multiple of direct losses is

sometimes used. However, when indirect loss is an important decision parameter, a great deal of research into

monetary evaluation may be necessary. Procedures for

valuing a human life and other indirect losses are discussed in Ramachandran.3

In the measurement of benefits, it is appropriate to

adjust for utility or disutility which may be associated

with a fire loss.

Costs may be divided into two major categories:

(1) costs of private fire protection services, and (2) costs of

public fire protection services. In either case, cost estimates will reflect the opportunity cost of providing the

service. For example, the cost of building a fire-resistive

structure is the production foregone due to the diversion

of labor and resources to make such a structure. Similarly,

the cost of a fire department is the loss of other community services which might have been provided with the

resources allocated to the fire department.

Selection of Best Alternative

There are two considerations in determining benefitcost criteria. The first pertains to project acceptability,

while the second pertains to project selection.

Project acceptability may be based on benefit-cost difference or benefit-cost ratio. Benefit-cost ratio is a measure

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Fire Risk Analysis

of project worth in which the monetary equivalent benefits are divided by the monetary equivalent costs. The first

criterion requires that the value of benefits less costs be

greater than zero, while the second criterion requires that

the benefit-cost ratio be greater than one.

The issue is more complicated in the case of project

selection, since several alternatives are involved. It is no

longer a question of determining the acceptability of a

single project, but rather selecting from among alternative

projects. Consideration should be given to changes in

costs 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 horizontal

axis, while the marginal costs and benefits associated

with various levels of fire safety are given on the vertical

axis. As the diagram indicates, marginal costs are low initially and then increase. Less information is available concerning the marginal benefit curve, and it may, in fact, be

horizontal. The economically optimum level of fire protection is given by the intersection of the marginal cost

and marginal benefit curves. Beyond this point, benefits

from increasing fire protection are exceeded by the costs

of providing the additional safety.

A numerical example is given in Table 5-7.3. There are

five possible strategies or programs possible. The first

strategy, A, represents the initial situation, while the remaining four strategies represent various fire loss reduction activities, each with various costs. Strategies are

arranged in ascending order of costs. Fire losses under

each of the five strategies are given in the second row,

while the sum of fire losses and fire reduction costs for

each strategy is given in the third row. The sum of fire

losses and fire reduction costs of each strategy is equivalent to the life-cycle cost of that strategy. Life-cycle cost

analysis is an alternative to benefit-cost analysis when the

outcomes of the investment decision are cost savings

rather than benefits per se. Additional information on

life-cycle cost analysis is found in Fuller and Petersen.5

Marginal costs

Table 5-7.3

Use of Benefit-Cost Analyses in Strategy

Selection

Strategy

Category

A

B

C

D

E

Fire reduction costs

Fire losses

Sum of fire reduction

costs and fire losses

0

100

10

70

25

50

45

40

70

35

100

80

75

85

105

0

0

30

10

20

15

10

20

5

25

0

—

20

3.0

5

1.33

–10

0.5

–20

0.2

Marginal benefits

Marginal costs

Marginal benefits minus

marginal costs

Marginal benefit-cost ratio

Data in the first two rows may then be used to determine the marginal costs or marginal benefits from the replacement of one strategy by another. Thus, strategy B

has a fire loss of $70 compared to $100 for strategy A, so

the marginal benefit is $30. Similarly, the marginal benefit

from strategy C is the reduction in fire losses from B to C

or $20. The associated marginal cost of strategy C is $15.

Declining marginal benefits and rising marginal costs result in the selection of strategy C as the optimum strategy.

At this point, the difference between marginal benefits

and marginal costs is still positive.

Marginal benefit-cost ratios are given in the last row.

It is worth noting that, while the highest marginal benefitcost ratio is reached at activity level B (as is the highest

marginal benefit-cost difference), project C is still optimum, since it yields an additional net benefit of $5. This

finding is reinforced by examining changes in the sum of

fire losses and fire reduction costs (i.e., life-cycle costs).

Total cost plus loss first declines, reaching a minimum at

point 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 and benefits, and minimizing the

sum of fire losses and fire reduction costs—yield identical

outcomes.

Treatment of Uncertainty

$

Marginal benefits

A final issue concerns the treatment of uncertainty.

One method for explicitly introducing risk considerations

is to treat benefits and costs as random variables which

may be described by probability distributions. For example, an estimate of fire losses might consider the following

events: no fire, minor fire, intermediate fire, and major fire.

Each event has a probability of occurrence and an associated damage loss. The total expected loss (EL) is given by

EL C

0

100%

Degree of fire safety

Figure 5-7.3.

Project selection.

3

}

piDi

iC0

where

p0 C probability of no fire

p1 C probability of a minor fire

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Engineering Economics

p2 C probability of an intermediate fire

p3 C probability of a major fire

Dn C associated damage loss, n C 0,1,2,3

Expected losses may be computed for different fire

protection strategies. Thus, a fire protection strategy that

costs C3 and reduces damage losses of a major fire from D3

to D3 will result in an expected loss

EL C p0D0 = p1D1 = p2D2 = p3D3 = C3

Similarly, a fire control strategy that costs C2 and reduces the probability of an intermediate fire from p2 to p 2

has an expected loss

EL C p0D0 = p1D1 = p 2D2 = p3D3 = C2

A comparison of expected losses from alternative

strategies may then be used to determine the optimal

strategy.

Use of expected value has a limitation in that only the

average value of the probability distribution is considered. Discussion of other procedures for evaluating uncertain outcomes is given by Anderson and Settle.6

References Cited

1. ASTM E833, Definitions of Terms Relating to Building Economics, American Society for Testing and Materials, West Conshohocken, PA (1999).

2. ASTM Standards on Building Economics, 4th ed., American Society for Testing and Materials, West Conshohocken, PA (1999).

3. G. Ramachandran, The Economics of Fire Protection, E & FN

Spon, London (1998).

4. American National Standards Institute Standard Z94.0-1982,

“Industrial Engineering Terminology,” Chapter 5, Engineering Economy, Industrial Engineering and Management Press,

Atlanta, GA (1983).

5. S.K. Fuller and S.R. Petersen, “Life-Cycle Costing Manual

for the Federal Energy Management Program,” NIST Handbook 135, National Institute of Standards and Technology,

Gaithersburg, MD (1996).

6. L.G. Anderson and R.E. Settle, Benefit-Cost Analysis: A Practical Guide, Lexington Books, Lexington, MA (1977).

Additional Readings

R.E. Chapman, “A Cost-Conscious Guide to Fire Safety in Health

Care Facilities,” NBSIR 82-2600, National Bureau of Standards, Washington, DC (1982).

R.E. Chapman and W.G. Hall, “Code Compliance at Lower

Costs: A Mathematical Programming Approach,” Fire Technology, 18, 1, pp. 77–89 (1982).

L.P. Clark, “A Life-Cycle Cost Analysis Methodology for Fire Protection Systems in New Health Care Facilities,” NBSIR 822558, National Bureau of Standards, Washington, DC (1982).

W.J. Fabrycky, G.J. Thuesen, and D. Verma, Economic Decision

Analysis, 3rd ed., Prentice Hall International, London (1998).

E.L. Grant, W.G. Ireson, and R.S. Leavenworth, Principles of Engineering Economy, 8th ed., John Wiley and Sons, New York

(1990).

J.S. McConnaughey, “An Economic Analysis of Building Code

Impacts: A Suggested Approach,” NBSIR 78-1528, National

Bureau of Standards, Washington, DC (1978).

D.G. Newnan and J.P. Lavelle, Engineering Economic Analysis, 7th

ed., Engineering Press, Austin, TX (1998).

C.S. Park, Contemporary Engineering Economics, 2nd ed., AddisonWesley, Menlo Park, CA (1997).

J.L. Riggs, D.D. Bedworth, and S.U. Randhawa, Engineering Economics, 4th ed., McGraw-Hill, New York (1996).

R.T. Ruegg and H.E. Marshall, Building Economics: Theory and

Practice, Van Nostrand Reinhold, New York (1990).

R.T. Ruegg and S.K. Fuller, “A Benefit-Cost Model of Residential

Fire Sprinkler Systems,” NBS Technical Note 1203, National

Bureau of Standards, Washington, DC (1984).

W.G. Sullivan, J.A. Bontadelli, and E.M. Wicks, Engineering Economy, 11th ed., Prentice Hall, Upper Saddle River, NJ (2000).

Appendix A: Symbols and Definitions of Economic Parameters

Symbol

Definition of Parameter

Aj

A

–

A

Cash flow at end of period j

End-of-period cash flows (or equivalent end-of-period values) in a uniform series continuing for a specified number of periods

Amount of money (or equivalent value) flowing continuously and uniformly during each period, continuing for a specified number

of periods

Future sum of money—the letter F implies future (or equivalent future value)

Uniform period-by-period increase or decrease in cash flows (or equivalent values); the arithmetic gradient

Number of compounding periods per interest perioda

Number of compounding periods

Present sum of money—the letter “P” implies present (or equivalent present value). Sometimes used to indicate initial capital

investment.

Salvage (residual) value of capital investment

Rate of price level increase or decrease per period; an “inflation” of “escalation” rate

Uniform rate of cash flow increase or decrease from period to period; the geometric gradient

Effective interest rate per interest perioda (discount rate), expressed as a percent or decimal fraction

Nominal interest rate per interest period,a expressed as a percent or decimal fraction

F

G

M

N

P

S

f

g

i

r

aNormally, but not always, the interest period is taken as 1 yr.

Subperiods, then, would be quarters, months, weeks, and so forth.

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Fire Risk Analysis

Appendix B: Functional Forms of Compound Interest Factorsa

Name of Factor

Algebraic

Formulation

Functional

Format

Group A. All cash flows discrete: end-of-period compounding

Compound amount (single payment)

(1 = i)N

(F/P, i, N)

Present worth (single payment)

(1 = i)–N

(P/F, i, N)

Sinking fund

i

(1 = i )N – 1

(A/F, i, N)

Capital recovery

i(1 = i)N

(1 = i )N – 1

(A/P, i, N)

Compound amount (uniform series)

(1 = i )N – 1

i

(F/A, i, N)

Present worth (uniform series)

(1 = i )N – 1

i(1 = i )N

(P/A, i, N)

Arithmetic gradient to uniform series

(1 = i )N – iN – 1

i (1 = i )N – i

(A/G, i, N)

Arithmetic gradient to present worth

(1 = i )N – iN – 1

i 2(1 = i )N

(P/G, i, N)

Continuous compounding compound amount

(single payment)

erN

(F/P, r, N)

Continuous compounding present worth (single payment)

e–rN

(P/F, r, N)

Continuous compounding present worth (single payment)

erN – 1

erN(er – 1)

(P/A, r, N)

Continuous compounding sinking fund

er – 1

erN – 1

(A/F, r, N)

Continuous compounding capital recovery

erN(er – 1)

erN – 1

(A/P, r, N)

Continuous compounding compound amount

(uniform series)

erN – 1

er – 1

(F/A, r, N)

Continuous compounding sinking fund (continuous,

uniform payments)

r

erN – 1

–

A /F, r, N

Continuous compounding capital recovery (continuous,

uniform payments)

rerN

erN – 1

–

A /P, r,N

Continuous compounding compound amount (continuous,

uniform payments)

erN – 1

r

–

F/A , r, N)

Continuous compounding present worth (continuous,

uniform payments)

erN – 1

rerN

–

P/A , r, N)

Group B. All cash flows discrete: continuous compounding

at nominal rate r per period

Group C. Continuous, uniform cash flows: continuous

compounding

aSee Appendix A for definitions of symbols used in this table.

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Engineering Economics

Appendix C—Interest Tables

Table C-7.1

Present Worth Factor (Changes F to P)

Yr.

2%

4%

6%

8%

10%

12%

15%

20%

25%

30%

40%

50%

1

2

3

4

5

.9804

.9612

.9423

.9238

.9057

.9615

.9246

.8890

.8548

.8219

.9434

.8900

.8396

.7921

.7473

.9259

.8573

.7938

.7350

.6806

.9091

.8264

.7513

.6830

.6209

.8929

.7972

.7118

.6355

.5674

.8696

.7561

.6575

.5718

.4972

.8333

.6944

.5787

.4823

.4019

.8000

.6400

.5120

.4096

.3277

.7692

.5917

.4552

.3501

.2693

.7143

.5102

.3644

.2603

.1859

.6667

.4444

.2963

.1975

.1317

6

7

8

9

10

.8880

.8706

.8535

.8368

.8203

.7903

.7599

.7307

.7026

.6756

.7050

.6651

.6274

.5919

.5584

.6302

.5835

.5403

.5002

.4632

.5645

.5132

.4665

.4241

.3855

.5066

.4523

.4039

.3606

.3220

.4323

.3759

.3269

.2843

.2472

.3349

.2791

.2326

.1938

.1615

.2621

.2097

.1678

.1342

.1074

.2072

.1594

.1226

.0943

.0725

.1328

.0949

.0678

.0484

.0346

.0878

.0585

.0390

.0260

.0173

11

12

13

14

15

.8043

.7885

.7730

.7579

.7430

.6496

.6246

.6006

.5775

.5553

.5268

.4970

.4688

.4423

.4173

.4289

.3971

.3677

.3405

.3152

.3505

.3186

.2897

.2633

.2394

.2875

.2567

.2292

.2046

.1827

.2149

.1869

.1625

.1413

.1229

.1346

.1122

.0935

.0779

.0649

.0859

.0687

.0550

.0440

.0352

.0558

.0429

.0330

.0254

.0195

.0247

.0176

.0126

.0090

.0064

.0116

.0077

.0051

.0034

.0023

16

17

18

19

20

.7284

.7142

.7002

.6864

.6730

.5339

.5134

.4936

.4746

.4564

.3936

.3714

.3503

.3305

.3118

.2919

.2703

.2502

.2317

.2145

.2176

.1978

.1799

.1635

.1486

.1631

.1456

.1300

.1161

.1037

.1069

.0929

.0808

.0703

.0611

.0541

.0451

.0376

.0313

.0261

.0281

.0225

.0180

.0144

.0115

.0150

.0116

.0089

.0068

.0053

.0046

.0033

.0023

.0017

.0012

.0015

.0010

.0007

.0005

.0003

21

22

23

24

25

.6698

.6468

.6342

.6217

.6095

.4388

.4220

.4057

.3901

.3751

.2942

.2775

.2618

.2470

.2330

.1987

.1839

.1703

.1577

.1460

.1351

.1228

.1117

.1015

.0923

.0926

.0826

.0738

.0659

.0588

.0531

.0462

.0402

.0349

.0304

.0217

.0181

.0151

.0126

.0105

.0092

.0074

.0059

.0047

.0038

.0040

.0031

.0024

.0018

.0014

.0009

.0006

.0004

.0003

.0002

30

35

40

45

50

.5521

.5000

.4529

.4102

.3715

.3083

.2534

.2083

.1712

.1407

.1741

.1301

.0972

.0727

.0543

.0994

.0676

.0460

.0313

.0213

.0573

.0356

.0221

.0137

.0085

.0334

.0189

.0107

.0061

.0035

.0151

.0075

.0037

.0019

.0009

.0042

.0017

.0007

.0003

.0001

.0012

.0004

.0001

60

70

80

90

100

.3048

.2500

.2051

.1683

.1380

.0951

.0642

.0434

.0293

.0198

.0303

.0169

.0095

.0053

.0030

.0099

.0046

.0021

.0010

.0005

.0033

.0013

.0005

.0002

.0011

.0004

.0002

1

(1 = i )y

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Fire Risk Analysis

Table C-7.2

Capital Recovery Factor (Changes P to A)

Yr.

2%

4%

6%

8%

10%

12%

15%

20%

25%

30%

40%

50%

1

2

3

4

5

1.020

.5150

.3468

.2626

.2122

1.040

.5302

.3603

.2755

.2246

1.060

.5454

.3741

.2886

.2374

1.080

.5608

.3880

.3019

.2505

1.100

.5762

.4021

.3155

.2638

1.120

.5917

.4163

.3292

.2774

1.150

.6151

.4380

.3503

.2983

1.200

.6545

.4747

.3863

.3344

1.250

.6944

.5123

.4234

.3719

1.300

.7348

.5506

.4616

.4106

1.400

.8167

.6294

.5408

.4914

1.500

.9000

.7105

.6231

.5758

6

7

8

9

10

.1785

.1545

.1365

.1225

.1113

.1908

.1666

.1485

.1345

.1233

.2034

.1791

.1610

.1470

.1359

.2163

.1921

.1740

.1601

.1490

.2296

.2054

.1874

.1736

.1627

.2432

.2191

.2013

.1877

.1770

.2642

.2404

.2229

.2096

.1993

.3007

.2774

.2606

.2481

.2385

.3388

.3163

.3004

.2888

.2801

.3784

.3569

.3419

.3312

.3235

.4613

.4419

.4291

.4203

.4143

.5481

.5311

.5203

.5134

.5088

11

12

13

14

15

.1022

.0946

.0881

.0826

.0778

.1141

.1066

.1001

.0947

.0899

.1268

.1193

.1130

.1076

.1030

.1401

.1327

.1265

.1213

.1168

.1540

.1468

.1408

.1357

.1315

.1684

.1614

.1557

.1509

.1469

.1911

.1845

.1791

.1747

.1710

.2311

.2253

.2206

.2169

.2139

.2735

.2685

.2645

.2615

.2591

.3177

.3135

.3102

.3078

.3060

.4101

.4072

.4051

.4036

.4026

.5059

.5039

.5026

.5017

.5011

16

17

18

19

20

.0737

.0700

.0667

.0638

.0611

.0858

.0822

.0790

.0761

.0736

.0990

.0954

.0924

.0896

.0872

.1130

.1096

.1067

.1041

.1019

.1278

.1247

.1219

.1195

.1175

.1434

.1405

.1379

.1358

.1339

.1679

.1654

.1632

.1613

.1598

.2114

.2094

.2078

.2065

.2054

.2572

.2558

.2546

.2537

.2529

.3046

.3035

.3027

.3021

.3016

.4019

.4013

.4009

.4007

.4005

.5008

.5005

.5003

.5002

.5002

21

22

23

24

25

.0588

.0566

.0547

.0529

.0512

.0713

.0692

.0673

.0656

.0640

.0850

.0830

.0813

.0797

.0782

.0998

.0980

.0964

.0950

.0937

.1156

.1140

.1126

.1113

.1102

.1322

.1308

.1296

.1285

.1275

.1584

.1573

.1563

.1554

.1547

.2044

.2037

.2031

.2025

.2021

.2523

.2519

.2515

.2512

.2510

.3012

.3009

.3007

.3006

.3004

.4003

.4002

.4002

.4001

.4001

.5000

30

35

40

45

50

.0446

.0400

.0366

.0339

.0318

.0578

.0536

.0505

.0483

.0466

.0726

.0690

.0664

.0647

.0634

.0888

.0858

.0839

.0826

.0817

.1061

.1037

.1023

.1014

.1009

.1241

.1223

.1213

.1207

.1204

.1523

.1511

.1506

.1503

.1501

.2008

.2003

.2001

.2001

.2000

.2503

.2501

.2500

.3001

.4000

60

70

80

90

100

.0288

.0267

.0252

.0241

.0232

.0442

.0428

.0418

.0412

.0408

.0619

.0610

.0606

.0603

.0602

.0808

.0804

.0802

.0801

.0800

.1003

.1001

.1000

.1200

.1500

i (1 = i )y

(1 = i )y – 1

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