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The Engineering Economist,51:347–359

Copyright?2006Institute of Industrial Engineers

ISSN:0013-791X print/1547-2701online

DOI:10.1080/00137910600987602

ECONOMICS OF ENGINEERING:A NEW FABRIC FROM

SOME OLD THREADS

Thomas W.Hill,Jr.

Industrial Engineering,St.Ambrose University,Davenport,Iowa,USA

Three axioms are used to present the basic fundamental ideas of engineer-ing economics.These axioms are then used to derive a generalization of present value that includes both a local and global perspective.The rela-tionship between the local and global concepts is derived and then used to solve cash?ow problems that involve?ows placed arbitrarily in time.

INTRODUCTION

Present value(present worth)is a widely used measure for comparing the economic value of alternatives.The investigation presented here is focused on the de?nition of present value,PV,a generalization of the concept,and its application in engineering economic analysis.The last“engineering economic”overview of PV was a survey of methods published in Remer et al.(1984).

Presentation of the concept of PV in modern engineering economic texts (e.g.,Blank and Tarquin,2005;Eschenbach,2003;Newnan et al.,2004; Park,2007;Sullivan et al.,2006;Thuesen and Fabrycky,2000;White et al., 1998)is generally accomplished in the following way.First,the context of economic analysis is established through examples and discussions.Then the fundamental ideas of interest rates and compound interest are intro-duced.These discussions are then combined to present the PV methodol-ogy.In this article,a different approach is taken.The fundamental ideas of engineering economics are presented using a set of axioms.The use of ax-ioms provides an ef?cient way to organize the basic ideas of a subject and is familiar to students.The idea of engineering economic axioms is also found in Stevens(2006).However,the axioms presented here are quite different in form.

Address correspondence to Thomas Hill,Jr.,St.Ambrose University,518W.Locust Street, Davenport,IA52803.E-mail:thomashill@http://www.wendangku.net/doc/1ba0da4e0b1c59eef8c7b4bf.html

348T.W.Hill,Jr.

Following a brief discussion,the axioms are used to derive the concepts of local and global PV.The relationship between local and global PV is then established using the axioms.This relationship is named the shifted?ow present value theorem.Next,the generalized ideas of PV are illustrated with two examples using cost?ow patterns that follow equations.Finally,the idea of delaying the start of the effect of a cash?ow sequence is investigated. This concept has been explored in Hill and Buck(1974)and Park and Sharp-Bette(1990).However,the present value formulas presented here are different because of the shifted?ow present value theorem.The use of these different present value formulas is illustrated with applications that use equations to represent patterns of cost?ows.

The ideas presented here were developed as a way to satisfy the fol-lowing objectives:1)Provide an alternative approach to introducing the fundamentals of engineering economics and the concept of PV that would increase the understanding of the material.2)Provide an increased number of analysis options for the practitioner.

AXIOMS OF ENGINEERING ECONOMIC ANALYSIS

In order to clearly and concisely present the fundamental principles of the economics of engineering,three axioms have been created and are presented below.A few basic de?nitions are needed to make the axiom statements concise.A time horizon is a sequence of time periods j(j=0, 1,2,...,n)over which a problem is to be analyzed.Let x=a reference period in time at which an analysis will be based.The interest rate to be used in the analysis is named i.The axioms of engineering economic analysis are:

The axiom of value:the value,V j,of a resource,at time j,is established in terms of a common monetary measure,usually dollars.

The axiom of time:the monetary value,V j,of a resource at time j is well represented at a different point in time,x,by the power function:

v(x,j)=V j(1+i)x?j(1) where j=0,1,2,...,n and x=an integer.

The axiom of equivalence:two alternative solutions are said to be equiv-alent if they have the same value when measured in the same way over the same time horizon using the same interest rate i.

Economics of Engineering349

The axiom of time has some immediate and familiar application.Speci?c names for equivalent?ows in period0(x=0)and period n(x=n)have long been established.They are a direct application of the time axiom:

P=v(0,j)=V j(1+i)?j,(2)

and F=v(n,j)=V j(1+i)n?j(3) where j is a period in the time horizon.

These values,P and F,have been used to compare monetary values of a system to each other at various points in time in a consistent way.The axiom of time also tells us that these ideas can be generalized as follows. Every calculated value v(x,j)is either a P to a given value V j if x

The axiom of equivalence has been stated very generally and thus some discussion is appropriate.Clearly,all alternatives must solve the same exact problem and be judged by the same rules.The implication of this for the choice of the time horizon is stated as follows.

Analysis Principle.The requirements of the problem establish the time horizon for all feasible alternatives.

Collectively,the axioms establish the framework for engineering eco-nomics.The intent is to bring the student to the following conclusion.The economic dynamics of an engineering activity(model,project,design,etc.) can be captured accurately by paying attention to these axioms whenever a problem is formulated.The importance and role of both physical and temporal resources are clearly included along with a measurement and comparison mechanism.

IMPLEMENTING EQUIVALENCE MEASURES

The axiom of equivalence tells us we must compare alternatives using exactly the same measure.From a practical perspective,the measure should be simple,easy to use,and consistent in its results.The time horizon of every problem has two endpoints or boundaries,the start(time0)and the end(time n).All alternatives are constrained by these boundaries.Let us call these endpoints present time(x=0in the time axiom)and future time(x=n in the time axiom).The equivalence axiom also requires a consistent way to measure all the economic activity occurring over the

350T.W.Hill,Jr. entire time horizon of a problem.The time axiom suggests a way to do this using either of the boundaries.

Suppose we choose the start(time0,x=0)as the reference http://www.wendangku.net/doc/1ba0da4e0b1c59eef8c7b4bf.htmling the time axiom we can compute the equivalent value of each?ow,V j,in the time horizon and then simply add them up to obtain a total equivalent value:

n

j=1

v(0,j)=V l(1+i)?1+V2(1+i)?2+···+V n(1+i)?n

=

n

j=1

V j(1+i)?j(4)

Let us call Equation(4)the global present value,PV.

In the same way the concept of a global future value,FV,can also be derived from the axioms:

FV=FV x=n=

n

j=1

V j(1+i)j(5)

When the V j follow a pattern that can be represented by an equation,the sum can be expressed as a closed form formula.These formulae may be simple enough so that the values for various i and n can be presented in a table.

Suppose the pattern does not begin at j=1and does not end at j=n. Rather the pattern starts at some intermediate period h+1and ends at another period k,where h+1

SHIFTED IN TIME

Suppose we have a series,S,of contiguous values(e.g.,a uniform or gradi-ent series or any series that follows a formula).Assume that the series starts at time period h+1and ends at period k.Following the above approach with x=h as the reference point yields

k

j=h+1v(h,j)=

k

j=h+1

V j(1+i)?(j?h)(6)

De?ne this value as the local present value of the series and denote it PV x=h(S)or PV h(S).

Economics of Engineering351 The local present value of a series,S,at period h can be determined using the present value formula for the series by recognizing that the duration of the series is k?h.This is an application of the time axiom.Traditional literature de?nes the duration of a collection of cash?ows as n.For the local present value set n=k?h.

SHIFTED FLOW PRESENT VALUE THEOREM.

If PV h(S)=local present value of series S(a pattern of contiguous?ows) starting at time h≥1and continuing through to time k>h+1,then the global present value at time0,PV,is

PV=PV h(S)(1+i)?h=

k

j=h+1

V j(1+i)?(j?h)(1+i)?h.(7)

P ROOF.The local present value for series S is de?ned by(6),and it can be expressed as follows:

PV h(S)=

k

j=h+1

V j(1+i)?(j?h).

In the axiom of time(1),set j=h,x=0,V h=PV h(S),then

v(0,h)=PV h(S)(1+i)?h=PV

and the proof is complete.

The above result gives us another way to view the ideas presented so far. Once computed,a local PV h can be thought of as just another V j in some larger context.Its meaning is drawn from the context of how it is viewed in the analysis.If seen from the perspective of the values that follow it,it is a local present value.If seen from the perspective of those values that precede it,it is a local future value.

LOCAL PRESENT VALUE FROM THE TRADITIONAL PERSPECTIVE

The idea of a local present value and its relationship to the global present value can also be determined by traditional means.Consider a series of ?ows,V j>0only for j=h+1,h+2,...,k.The global present value

352T.W.Hill,Jr. of this series is by de?nition

PV=

k

j=1

V j(1+i)?j(8)

Note that V j=0for j

PV=

k

j=h+1

V j(1+i)?(j?h)(1+i)?h=(1+i)?h

k

j=h+1

V j(1+i)?(j?h)(9)

The same result as given the shifted?ows theorem(7).

The shifted?ow theorem along with the axioms will be used in the development of the ideas in the remainder of the article. APPLICATION OF THE SHIFTED FLOW PRESENT

VALUE THEOREM

One advantage of the shifted?ow theorem(7)is that we can use standard engineering formulas based on physical considerations to represent future cash?ows located anywhere in the time horizon.The formulas can then be used to investigate the economic aspects of the engineering design of a product or system.The standard formulas for the uniform series and gra-dient series are immediately applicable.Virtually any series of cash?ows represented by a formula can be incorporated into the traditional engineer-ing economics toolbox of present values.The present value formulas for some common engineering functions are given in Table1.It is important to note that these present value formulas start in period one just as the common formulas for the uniform and arithmetic series.Note that the present value formulas for the uniform and arithmetic gradient series have been included in Table1so that the added formulas can be related to the traditional ones. EXAMPLE USES OF TABLE1AND SHIFTED FLOW PRESENT VALUE THEOREM

To illustrate the use of the ideas presented,consider the following situation. An individual needs to set up an annuity to draw on starting seven years from now.It has been determined that the amount needed at the end of year seven is at least$2500,and that need will grow geometrically at5%over the following three years.Due to current?nancial con?icts,no deposits can be made until the end of year two.In addition,it is anticipated that income will grow geometrically at a rate of3%over the years we are

Economics of Engineering353 Table1.Present value formulas for non-delayed cost functions

Series name Function Present value

Exponential decay V j=Ce?jr j=1,

2,3,...,n

Ce?r

1+i?e?r

{1?e?nr

(1+i)n

}

Growth V j=C[1?e?jr]

j=1,2,3,...,n C{1

i

?1

i(1+i)n

?e?r?r[1?e?nr

(1+i)n

]}

Geometric V j=Ca j j=1,2,

3,...,n

Ca

1+i?a

[1?a n

(1+i)n

]

Uniform V j=C j=1,2,

3,...,n C[1

i

?1

i(1+i)n

]

Arithmetic gradient V j=C+G(j?1)

j=1,2,3,...,n (C+G

i

)[1

i

?1

i(1+i)n

]?Gn

i(1+i)n

paying into the annuity.It is desired to keep the payment schedule in a constant relationship with the ability to pay.If the interest rate is6%per year,what is the?ve-year schedule of deposits for years two through six?

The payment plan variable and the local present value can be determined using the geometric series function in Table1as follows.First,the value of C must be determined from the equation and the given need.Thus, V7=2500=C(1.05)and

C=2500/1.05=2380.95(10) To assure the$2500desired for year seven set C=2380.96.

The local present value(PV from Table1)needed to support the desired withdrawals is

PV6=2380.96(1.05)

(1.06?1.05)

1?

(1.05)4

(1.06)4

=$9301.33.(11)

The time axiom tells us this value is also a local FV for those?ows that precede it.In this situation,x=1,j=6,V6=9301.33,and

v(1,6)=V6(1.06)(x?j)=9301.33(1.06)?5=6950.49(12) However,v(1,6)is also the local present value for the?ows that follow it;

i.e.,it is PV1.Again,using Table1the present value is

PV1=6950.49=

C(1.03)

(1.06?1.03)

1?

(1.03)5

(1.06)5

(13)

354T.W.Hill,Jr. and

C=

(.03)6950.49

1.03

1?(1.03)5

(1.06)5

=1513.89(14)

From the geometric function in Table1the year end deposit schedule is: V2=1559.30;V3=1606.08;V4=1654.26;V5=1703.90;V6=1755.01 Note that only local PVs were required to solve this problem.

As a second example,consider the following situation.Over the next 24periods two separate cash?ow sequences will take place.We are in-terested in the impact of each.The?rst sequence starts in period3and continues through period10.The values of the sequence are determined by the exponential decay function with a constant multiplier of10,000and an exponent value of.085.The second sequence starts in period16and runs through period24.The values of the second sequence are determined by the geometric growth function with constant value5,000and multiplier value of1.03.What is the overall present value if the interest rate is10%?

To use Table1the value of n must be determined for each sequence.For the?rst sequence,n=k–h=10–2.Thus,

PV2=

1000e?.085

(1.1?e)

1?

e?8(.085)

(1.1)8

=38,648.93(15)

For the second sequence,n=24–15=9,and

PV15=5000(1.03)

(1.1?1.03)

1?

(1.03)9

(1.1)9

=32,860.59(16)

Finally,the global present value is

PV=PV2(1.1)?2+PV15(1.1)?15

=31,941.26+7,866.56=39,807.82(17) The size of the local present values as well as their impact on the global present value is clearly illustrated.

All of these same ideas can be applied to future values as well.In either case a local PV or FV can be seen as nothing more than a?ow value V j at time j.

Economics of Engineering355 SHIFTED CASH FLOWS WITH DELAYED START

Suppose the V j are determined by an equation,but the initial values are to be ignored.The shifted?ows theorem permits delayed start or truncated cash?ows to be treated the same as any other.The delayed start is thus presumed to begin in period one just as before.

Consider the following situation.We have purchased a new process and it will be started operating now.The formula predicting the maintenance costs is known.However,the warranty agreement is such that we will experience no maintenance cost for d periods.For economic analysis we need to delay the impact of maintenance cost?ows for d periods,while still allowing the value of the?ows to follow the established formula for the process behavior.

The general methods followed in Hill and Buck(1974)and Park and Sharp-Bette(1990)will be used to derive the appropriate present value formula that follows the shifting philosophy already presented.The intent is to demonstrate the approach so that any engineer could derive the formula needed for their particular application.

Graphically,delaying the start of a sequence of?ows is equivalent to moving the graph of the formula left on the x1–N axis.Algebraically, delaying the start of the sequence is equivalent to adding d to the indexing parameter of the formula.For example,assume the delay will be d periods and the?ows follow the exponential decay formula.Then

V j=Ce?(j+d)r;j=0,1,...,k(18) Recall that,we always start the global present value calculation of a se-quence at j=1.Following the references we rewrite V j as

V j=f(j)=C{u(j?1)?u[j?(k+1)]}e?(j+d)r(19) The unit step function1is used to start and stop the?ows so as to cover the desired periods.Looking at the?rst portion of the function involving u(j?1),call it f1(j)and substitute into the de?nition of the Z transform, thus

Z{f1(j)}=

∞

j=0

f1(j)z?j=

∞

j=0

C[u(j?1)e?(j+d)r]z?j.(20)

1The unit step function is de?ned as follows:u(j?a)={1,j≥a;0,otherwise}.

356T.W.Hill,Jr. Let j–1=n or(n=j+1),then

Z{f1(j)}=

∞

n=?1

u(n)Ce?(n+1+d)r z?(n+1)(21)

Since u(n)=0f or n<0we have

Z{f1(j)}=Ce?(d+1)r z?1

∞

n=0

e?nr z?n(22)

Using any table of Z transform pairs(Park and Sharp-Bette,1990;Zwill-inger,2002)we?nd

Z{f1(j)}=Ce?(d+1)r z?1

z

(z?e)

(23)

Letting z=1+i and simplifying

PV{f1(j)}=

Ce?(d+1)r

(1+i?e)

(24)

Following these same steps we can derive the present value expression for f2(j).When these two expressions are combined the present value for the exponential decay function delayed by d periods is as follows.

PV{f(j)}=

Ce?(d+1)r

(1+i?e?r)

1?

e?nr

(1+i)n

(25)

This present value formula along with those for growth and geometric growth are shown in Table2.It is important to note that for the delayed ?ows n=k–h–d.

EXAMPLE OF DELAYED STARTING AND USE OF TABLE2 WITH THE SHIFTED FLOWS THEOREM

Two processes are being considered to satisfy a need.They are technically equal but have very different operating costs and some different purchase options are being offered by the suppliers.The chosen process will be started at the end of period0and will run for9periods.The operating costs for process one are described by an exponential decay equation with a constant value of10,000and an exponent value of.085.The provider is offering to pay the?rst years operating costs as an inducement to buy. The operating costs for process two are described by a geometric growth

Economics of Engineering357 Table2.Present value formulas for delayed start cost functions

Series name Function Present value

Exponential decay V j=Ce?(j+d)r

j=1,2,3,...,n Ce?(d+1)r

?r

[1?e?nr

(1+i)n

]

Growth V j=C[1–e?(j+d)r]

j=1,2,3,...,n C{1

i

?1

i(1+i)n

?e?(d+1)r

1+i?e?r

[1?e?nr

(1+i)n

]}

Geometric V j=Ca j+d j=1,

2,3,...,n Ca(d+1)

1+i?a

[1?a n

(1+i)n

]

equation with a constant value of6,000and a multiplier value of1.03.The manufacturer is offering to pay the operating costs for the?rst two years. The interest rate for evaluating alternatives is10%,which process should be recommended?

To use Table2,the proper values of n and d must be established for each alternative.For process one,d=1,h=0,k=9,so n=9?0?1=8. Thus,

PV1=10,000e?2(.085)

(1.1?e?.085)

1?

e?8(.085)

(1.1)8

=35,499.52(26)

and

PV=35,499.52(1.1)?1=32,272.29(27) Note,in general the subscript on the local PV for delayed and shifted?ows is h+d.

For process two,d=2,h=0,k=9,and n=7.Hence,

PV2=6,000(1.03)3

(1.1?1.03)

1?

(1.03)7

(1.1)7

=34,550.18(28)

and

PV=34,550.18(1.1)?2=28,553.87(29) Based on present value with i=.1,and the axiom of equivalence,process two is clearly recommended.

How important was it that the provider of process two offered to pay the second year of operating cost?To answer this set d=1,so n=8,

PV1=6,000(1.03)2

(1.1?1.03)

1?

(1.03)8

(1.1)8

=37,195.98(30)

358T.W.Hill,Jr. and

PV=37,195.98(1.1)?1=33,814.53(31) Since this new PV is larger than the competitor,the offer was the winning factor.

SUMMARY AND CONCLUSIONS

In an effort to present the ideas of engineering economics in a briefer way and more available to engineers in general,a number of ideas have been brought together in this article to create a different perspective of the fundamentals.The basics of engineering economics were presented in terms of three axioms.The axioms were then applied to generalize the concept of present value to include both a local and global view.Present value formulas were developed for various basic functions common to engineering analysis.The form of these present value formulas is based on the assumption that the?rst?ow occurs in period one,which is consistent with the traditional approach.

The power of all these ideas does not come from being able to work the problems by hand as was illustrated here.But,rather that the shifting and delay principle make it a straightforward matter to solve real problems using macros that can be readily applied in a spreadsheet to perform all the calculations.

If we accept the ideas presented here,we do not have to change the printed tables of factors.However,we will need to create an explanation of how to use/interpret the tables in light of the local PV perspective presented. Ps as traditionally de?ned in all the factors need to be recognized as a local present value(similarly for F).This could necessitate a much earlier formal de?nition of present value,at least in the local form.Global present value can be presented however the author wishes,perhaps when the concept of net present value is introduced.

The results here are consistent and compatible with those obtained pre-viously.Thus,the hope is that this approach will both widen the use and appreciation of engineering economics and facilitate a deepening in the understanding of the results as a consequence of the concepts presented here.

ACKNOWLEDGEMENT

Comments and questions from the reviewers have served to signi?cantly improve the presentation of the ideas in this article.

Economics of Engineering359 REFERENCES

Blank,L.T.,and Tarquin,A.(2005)Engineering economy,6th ed.McGraw-Hill Higher Education:New Y ork.

Eschenbach,T.(2003)Engineering economy:applying theory to practice.Oxford Uni-versity Press:New Y ork.

Hill,T.W.,Jr.,and Buck,J.R.(1974)Zeta transforms,present value,and economic analysis.AIIE Transactions,6(2),120–125.

Newnan,D.G.,Eschenbach,T.G.,and Lavelle,J.P.(2004)Engineering economic anal-ysis.Oxford University Press:New Y ork.

Park,C.S.(2007)Contemporary engineering economics,4th ed.Pearson Prentice Hall: Upper Saddle River,NJ.

Park,C.S.,and Sharp-Bette,G.P.(1990)Advanced engineering economics.Wiley:New Y ork.

Remer,S.D.,Tu,J.C.,Carson,D.E.,and Ganiy,S.A.(1984)The state of the art of present worth analysis of cash?ow distributions.Engineering Costs and Production Economics,7(4),257–278.

Stevens,A.M.(2006)The main things you should learn about the time value of money.

Available at http://www.unb.ca/web/transpo/mynet/mtw3.htm(retrieved15June 2005).

Sullivan,W.G.G.,Wicks,E.M.,and Luxhoj,J.(2006)Engineering economy,13th ed.

Prentice Hall:Upper Saddle River,NJ.

Thuesen,G.J.,and Fabrycky,W.J.(2000)Engineering Economy,9th ed.Prentice Hall: Upper Saddle River,NJ.

White,J.A.,Case,K.E.,Pratt,D.B.,and Agee,M.H.(1998)Principles of engineering economic analysis,4th ed.Wiley:New Y ork.

Zwillinger,D.(2002)Standard mathematical tables,31st ed.Chapman-Hall/CRC: Boca Raton,FL.

BIOGRAPHY

T HOMAS W.H ILL,J R graduated from Arizona State University in1962.After working as a design engineer in the aerospace industry,he returned to ASU where he earned a Ph.D.in industrial engineering in1969.He taught industrial engineering at Purdue University for 6years before returning to the aerospace industry in Phoenix.He spent the next20years in various positions as engineer and manager in both manufacturing and engineering.He joined the industrial engineering department at St.Ambrose University in1995.He has authored or coauthored more than20papers on various topics in engineering economics, operations research,and systems engineering.

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