Designing Pm

INTERNATIONAL JOURNAL OF

PROJECT MANAGEMENT International Journal of Project Management 24 (2006) 136–144 www.elsevier.com/locate/ijproman

Designing project management: A scientific notation and an improved formalism for earned value calculations Denis F. Cioffi

*

Project Management Program, The George Washington University, Monroe Hall 302, 2115 G Street NW, Washington, DC 20052, United States Received 5 August 2004; received in revised form 15 October 2004; accepted 15 July 2005

Abstract A new formalism and a corresponding new notation for earned value analysis are presented. With compact, consistent, mnemonic notation, earned value calculations become more transparent and flexible, leading to insights about standard quantities and advances through new measures. As an example of the notationÕs utility, it is used to generate a modified earned value approach that weights quantities according to their position in a projectÕs timeline.
2005 Elsevier Ltd and IPMA. All rights reserved. Keywords: Earned value; Managing projects; Progress; Cost; Cash flow management

1. Introduction: design and revolution Thomas Kuhn introduced the now oft-misused term ‘‘paradigm shift’’ in his seminal book, The Structure of Scientific Revolutions [1]. Scientists build models or ‘‘paradigms’’ to explain the composition and behaviour (physical and biological) of the universe. Most models work well when first proffered. With time, however, improved technology and more sophisticated experimental techniques lead to more and better data. Deeper thinking with these new data eventually shows shortcomings, inconsistencies, or downright errors in the accepted paradigm of the day. At first, the community responds with minor reconstructions of the paradigm so that it agrees with the new data and understanding. Eventually, however, the new perspective of the world differs so greatly from that provided through the old paradigm that a shift occurs. The old paradigm is replaced by a new one, e.g., special relativity replaces Newtonian dynamics *

Tel.: +1 202 994 9533; fax: +1 202 994 2736. E-mail address: denis.cioffi@gwu.edu.

0263-7863/$30.00
2005 Elsevier Ltd and IPMA. All rights reserved. doi:10.1016/j.ijproman.2005.07.003

at speeds near that of light. Such new models give new perspectives that provide better explanations of the world in which we live. In a manner analogous to shifts in scientific theories, earned value analysis began, we hope, a paradigm shift in project management. The actual progress of any given project has not changed, but our measurement techniques have changed because of a new perspective. Separate views of budget or schedule should not be accepted. Instead, good management demands the integrated view provided by earned value analysis. However, whatever their ultimate worth, paradigm shifts do not occur overnight, either in the physical sciences or in the Sciences of the Artificial [2], which include project management. In these sciences, according to Simon, ‘‘Everyone designs who devises courses of actions aimed at changing existing situations into preferred ones. . .’’ Just as earned value itself is a design aimed at changing the estimation of project progress into a ‘‘preferred situation’’, this contribution of a new notation for earned value is a design that aims to further the ongoing paradigm shift.

D.F. Cioffi / International Journal of Project Management 24 (2006) 136–144

2. Earned value ‘‘If you canÕt measure it, you canÕt manage it’’. Whether one trusts the validity of this common phrase most of the time or all of the time, measuring the true progress of a project presents a formidable task. Given a baseline plan, projects typically report a measure of the completed work and compare it to that scheduled. Similarly, most projects can and do measure the current cost and compare it to the planned spending. But for a more comprehensive view, how does one measure the progress of a project against the triple constraint of cost, schedule, and scope? The two simple measures above separate schedule and cost and include scope only indirectly, as a function of schedule. Post-World War II military projects advanced the field of project management. In 1967 the US Department of Defense released its first official list of ‘‘Cost/ Schedule Control Systems Criteria’’ (C/SCSC) [3], signaling the formal initiation of earned value analysis, which still represents managementÕs best chance at measuring a projectÕs progress in an integrated manner. Many (probably most) projects do not use earned value, however, and the historically arcane terminology and calculational notation have stood as roadblocks to its embrace by the management community. In an attempt to evolve the system to a more scientific format, this paper further formalizes a notation developed originally in Managing Project Integration [4]. 2.1. Efforts and S-curves Earned value analysis combines the three elements of budget, schedule, and scope by using cost as the common exchange medium. Thus, the unit of a projectÕs primary financial currency (e.g., dollars, pounds, the Euro) becomes the unit for all earned value measures. One can therefore compare different measurements because they have a common basis. How is this process possible? At least one published version of the standard project management S-curve plots ‘‘labor hours’’ to illustrate the rise and fall of effort through the life of a project. Adding labor hours generated by different resources, however, is the project management equivalent of adding those proverbial apples and oranges, e.g., adding the labor hours of the attorney and the bulldozer (or the bulldozer operator) makes no sense: the number of personnel can be added, but disparate efforts cannot be combined. Earned value avoids this problem by reducing efforts to a common basis—costs—and measuring those costs in a common unit of currency. To formalize the implicit assumptions underlying earned value, we can define the effort, ER, that results from a resource used at some intensity, RI, through a given duration, Dt. When considering the optimum duration of a task [4], the duration

137

and therefore the effort become functions of the resource intensity itself, but here we will use the common linear approximation ER
RI Dt.

ð1Þ

To quantify the example offered above, consider the resources attorneys and bulldozers in intensities of 1 and 10, respectively, for durations of 6 min and 2 days. The resulting efforts are six attorney minutes and 20 bulldozer days. One compares these efforts by converting them to costs through a cost rate, C_ R , that might be obtained from a resource breakdown structure [5]. The units of these particular cost rates are currency units per unit of effort (e.g., dollars per attorney minute), and the subscript R reminds us that the cost rate varies with each individual resource. To summarize C ¼ ER C_ R ;

ð2Þ

which properly introduces the generic cost, C, that becomes specific in earned value calculations. 2.2. Earned value defined The earned value system incorporates scope and integrates it with cost and schedule. First, the manager determines the value of a projectÕs fully completed or partially completed efforts (consistent with the effort definition above) in the context of the cost that was budgeted and (presumably) agreed upon in the project plans. Only when a specified amount of task work is accomplished does a project earn value, and the amount of that value is determined by the cost that was budgeted. I have seen no better definition and no better or more concise way to express this special type of value than ‘‘the budgeted cost of work performed’’, which in a shorthand we refer to as ‘‘earned value’’. In this framework, the mere contemplation of the budgeted cost of work performed cries out for an immediate comparison to the actual cost. Earned value analysis next brings the schedule into this common comparison basis by asking how much spending should have occurred, i.e., according to a projectÕs schedule, at the specific time of any comparison. 2.3. Standard notation Prior to the latest edition of the Project Management InstituteÕs A Guide to the Project Management Body of Knowledge (PMBOK) [6], the abbreviations used for these three essential earned value quantities were taken directly from the initials of the defining nomenclature:  BCWP: the budgeted cost of work performed  ACWP: the actual cost of work performed  BCWS: the budgeted cost of work scheduled

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Students (and some practitioners) would grind through calculations with these sets of initials. Confusion resulted. As Frame put it 10 years ago, ‘‘students spend more time trying to master the vocabulary than the concepts’’ [7]. (My own observations as an instructor agree with this perspective.) Moreover, in a recent review of the second edition of Earned Value Project Management [3], Kenneth Rose notes earned valueÕs ‘‘historically arcane and ponderous terminology’’ [8]. In an attempt at improving this terminology, the latest edition of the PMBOK reduces the number of words to two per cost term:  EV: Earned value (BCWP)  AC: Actual cost (ACWP)  PV: Planned value (BCWS) However, the removal of some key words has also removed information because ‘‘Value is an inherently subjective concept. To study value, it is necessary to first operationalize it in terms of variables that can be directly observed’’ [9]. (If you cannot measure it, you cannot manage it.) If asked what is meant by earned value, one could only reply, ‘‘the budgeted cost of work performed’’, for those words define the concept as used in project management—‘‘value’’ conveys its specific meaning only with this complete definition. Similarly, the ‘‘planned value’’ would be defined as ‘‘the budgeted cost of work scheduled’’. Only ‘‘actual cost’’ remains unambiguous. Two different issues have been confused. The words used in the ‘‘ponderous’’ definitions are not the problem. The problem lies in assuming that the terminology and the definitions should have a one-to-one relationship with the notation used for calculations. So, while the newest official terminology can be viewed as a step in the right direction, in the end little has been gained, and I propose a new formalism.

3. A new formalism Project managementÕs use of ‘‘value’’ in earned value qualifies it as a ‘‘term of art’’. A term of art is a word or phrase that takes on a special meaning in a particular discipline (notably law). These special meanings generally align with the wordsÕ standard meanings, which is why they are chosen, but the non-specialist will not appreciate the nuances that the word or term carries within the discipline. (Examples from physics include ‘‘force’’ and ‘‘velocity’’, and from the law, ‘‘assault’’.) The academics and the practitioners in the field have added these meanings. We can carry these additional meanings to a new notation. In fact, one of the definitions of ‘‘formalism’’

refers to ‘‘manipulation according to certain rules of intrinsically meaningless symbols’’ [10], which explains why the terminology and the notation need not be identical: just as we had to operationalize ‘‘value’’ to transform it to a project management term of art, we can give meaning to symbols created to illuminate calculations instead of obfuscating them. What other goals might we set for a new formalism? (1) The formalism should be consistent. As much as possible, the same symbol should refer to the same quantity or operation whenever it is used, and that quantity should always be referred to by the same basic symbol. (2) The notation should be mnemonic. Although we cannot expect the literal replication of the words used to define the earned-value concepts, the symbols should suggest the right words intrinsically. (3) Quantities with identical dimensions should have that commonality represented clearly so that one recognizes immediately a dimensionless result in the ratio of two such quantities, e.g., in one cost divided by another. (4) The notation should be relatively compact. Compact notation facilitates manipulation and understanding. Over the past several hundred years, the physical sciences and the mathematics they require have fostered the development of sophisticated symbolic schemes designed to convey a maximum of information with a minimum of notation. We should follow that lead. At first some additional effort must be put forth to understand new symbols that will not replicate the definitions, but the resulting ease in future manipulations justifies the initial energy expenditure. Using a (mathematical font) C as the common variable for the three primary earned-value costs sets us on the right path to meet the above objectives. We can distinguish the three Cs with appropriate subscripts. For the earned value itself, a subscript ‘‘b’’ reminds us of the ‘‘budgeted’’ cost; ‘‘work performed’’ is implicit here. The subscript ‘‘a’’ gives us ‘‘actual’’ directly, with ‘‘work performed’’ again implicit. I prefer ‘‘s’’ for a ‘‘scheduled’’ cost instead of ‘‘planned’’. At the start of a project, almost everything is—or should be—planned. A ‘‘scheduled’’ cost implies a ‘‘budgeted’’ cost, but it would not be that of the work performed because one does not schedule costs that have already occurred, i.e., actual ones. The listing below shows the evolution of the tripletÕs notation:  Earned value: BCWP ! EV ! Cb  Planned value: BCWS ! PV ! Cs  Actual cost: ACWP ! AC ! Ca

D.F. Cioffi / International Journal of Project Management 24 (2006) 136–144

139

Table 1 Notation correspondence Defined quantity

Notation

Eq. no.

(Terminology)

Standard

New

Budgeted cost of work performed (Earned value) Budgeted cost of work scheduled (Planned value) Actual cost of work performed (Actual cost) Difference between earned value and actual cost (Cost variance) Difference between earned value and scheduled cost (Schedule variance) Cost performance index {CPI(e)} Cost performance factor (Cost performance index) Schedule performance index Schedule performance factor Total project cost, as originally planned (Budget at completion) New estimate of actual project cost (Low-end cumulative CPI estimate at completion)

BCWP EV BCWS PV ACWP AC

Cb

(3)

Cs

(3)

Ca

(3)

DCa

(3)

DCs

(3)

Ic Fa

(4)

Is Fs Cs,1

(4) (9)

C 0a;1

(9)

CV SV CPI 1/CPI CPI(p) SPI 1/SPI BAC EAC

The correspondence between the standard notation and the proposed notation is shown. Although the new formalism prefers cost and schedule factors, the table also suggests a single letter (I) with the appropriate subscript for the standard cost and performance indices. The number in the right-hand column gives the first equation in which the symbol appears. The equation referenced may show the subscript ‘‘i’’ instead of either a (actual) or s (scheduled) because i can take on either of those values.

If one did not know the above history, would the new notation make sense? The Cs communicate ‘‘costs’’, and thus the notation indicates the common dimension of these quantities almost instantly. With the subscripts, one would read ‘‘budgeted cost’’, ‘‘scheduled cost’’, and ‘‘actual cost’’. Inquiring about the difference between ‘‘budgeted’’ and ‘‘scheduled’’, the novice would be told that the earned value system is designed

Cost (Normalized)

Fraction of Planned Total (Cs,1)

1

0.8

Earned value cost and schedule differences are measured vertically.

to measure progress and one must differentiate between work as scheduled and work as performed: we measure the value of the work performed in terms of its budgeted cost. Table 1 shows the correspondence between the standard notation and this proposed notation. Fig. 1 illustrates these fundamental earned value parameters on a cost versus time curve (an S-curve) where the plotted values have been normalized to the projectÕs total cost and total duration, respectively. In this example, the actual cost and the budgeted cost have been measured at 25% of the projectÕs planned total duration; as shown, the earned value (Cb) is less than both the scheduled (Cs) and actual (Ca) costs. The project team will discover that its project is behind schedule and over budget.

0.6

C

4. A scientific notation

s

0.4

Ca

0.2

In this section, the earned-value triplet parameters are combined to produce the standard derived quantities of earned value analysis. I shall also introduce a new quantity.

∆Ci =C b -C i

C

0

b

0

0.2

4.1. Differences 0.4

0.6

0.8

1

Time (Normalized) Fraction of Planned Total (T1)

Fig. 1. The earned value triplet on a normalized S-curve.

The differences between the budgeted cost and the two other costs tell whether a project is on, ahead of, or behind budget or schedule. These differences traditionally are referred to as ‘‘variances’’, presumably because the actual

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project varies from its plans. However, ‘‘variances’’— implying some statistical significance—promises more than it can deliver. ‘‘Difference’’ is the proper word for the quantity produced when one number is subtracted from another. We use the Greek letter D for the differences, and in an economy of notation both are presented here via a single equation: DC i
C b  C i ;

ð3Þ

where ‘‘i’’ becomes ‘‘a’’ to produce the cost difference and ‘‘s’’ to produce the schedule difference. This compact notation saves space and time when writing and viewing these quantities. More important, it emphasizes their sibling nature and discriminates between them only when necessary. When the quantity DCi is negative, a project is over budget (i = a) or behind schedule (i = s). Fig. 1 shows how one would measure these differences on a typical S-curve plot that shows normalized costs—actual, budgeted (of work performed), and scheduled—versus normalized time. (The curves were generated from the closed form presented by Cioffi [11].) 4.2. Cost and schedule factors One forms the standard earned value performance indices by comparing the earned value to each of the other two costs in the triplet. The cost performance index, CPI, is sometimes written CPI(e), with the ‘‘e’’ standing for ‘‘efficiency’’ to contrast with the inverse ratio, CPI(p), where ‘‘p’’ again stands for ‘‘performance’’[3] (see Table 1). Inverting both schedule and cost indices from their standard ‘‘efficiency’’ forms (suggested some years ago by Anbari [12]) and renaming them ‘‘factors’’ yields two advantages: (1) Both the actual cost and the scheduled cost are compared to the same quantity: the budgeted cost. Because they have a common denominator, they can be added together directly, suggesting the possibility of a new, combined parameter (one example will be shown shortly) that examines both cost and schedule together. (2) The ratio shows immediately the fractional differences between planned and actual cost and schedule. Again a single equation defines both factors Fi


Ci Cb

¼1

ð4aÞ DC i . Cb

ð4bÞ

When i = a, we have the cost factor, and when i = s we have the schedule factor. A project is on schedule and budget when Fa = Fs = 1 identically. Unlike the standard indices, a project is ahead of budget or schedule

when the factors are less than one. Users can decide the worth of this price of inversion by its compensating multiplicative advantage: a project is simply 100  ðF i  1Þ

ð5aÞ

percent over budget or behind schedule. (If Fi < 1 a project is ahead of budget or schedule, and the above difference is negative.) This expression can also be written as
 DC i 100  . ð5bÞ Cb The ratio DCi ‚ Cb gives the difference between the budgeted cost and the actual cost (i = a) or the scheduled cost (i = s) as a fraction of the budgeted cost (Cb). Any originally planned duration or cost will show a fractional change given by the appropriate Fi  1, and that combination will appear repeatedly when we use earned value for predictions, in the section ‘‘A Linear Future?’’ 4.3. A combined factor To see how the health of various projects compares to their original plans (and to each other), an executive in charge of more than a few projects wants to examine a minimum number of indicators: The minimum number would be one indicator per project, and I suggest the possibility of the following combination for a costschedule performance factor: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 U
Fa þ Fs  1 þ ðF a  F s Þ . ð6Þ 2 The subtraction of 1 allows the combined factor to equal 1 (as do the individual factors) if a project is both on schedule and on budget. The square root of the difference between the factors, squared, adds to the sum so that the effect of one high individual factor cannot be hidden by a low number in the other factor. For example, if Fa = 3/4 and Fs = 4/3, so that Fa + Fs  1  1, the square root term adds 0.41 so that U = 1.5, which highlights quickly the problem waiting to be discovered in the projectÕs schedule (the discrepancy between the factors may also be problematic); Table 2 collects the notation that has no standard earned value analog. Some earned value advocates maintain that the product of the standard cost and schedule indices, called the ‘‘Critical Ratio’’ [13], creates a good combined factor, but in the example just provided, that product is 4/3 · 3/4 = 1. A critical ratio of 1 says everything is proceeding according to plan, but most executives would want to know if one of their projects was 33% behind schedule (Fs  1 = 0.33); when glancing down a single column of numbers, U = 1.5 would alert them quickly.

D.F. Cioffi / International Journal of Project Management 24 (2006) 136–144 Table 2 Other defined quantities Defined quantity

Notation

Eq. no.

Resource intensity Effort from specific resource Duration of an effort Cost rate Variable cost or factor subscript Project duration as originally planned New estimate of project duration Difference between new and original project duration estimate Difference between new and original project cost estimate Remaining project cost Total budgeted cost in terms of current earned value Aggregate factors Time-weighted aggregate factors Weighting factor Cost-schedule performance factor

RI ER Dt C_ R i DT1 DT 01 dT1

(1) (1) (1) (2) (3) (7) (7) (8a)

dC1

(3)

dCa Fs,1

(10a) (10b)

F
i hF
i i wj
U; U

(15) (21) (17) (6)

The notation used for quantities that are newly defined, are not specific to earned value, or are not typically given separate notation in the standard treatment is shown. The number in the right-hand column gives the first equation in which the symbol appears.

5. A linear future? In its focus on developing a new notation, this paper accepts the current standard earned value assumption of a linear extrapolation of past trends to predict project performance. With this assumption, the cost and schedule factors (initially defined in Eqs. (4)) describe not just the past of a project, but its future, too. If earned value measurements show that a project is 15% behind schedule (Eq. 5a: 100 · [Fs  1] = 15) and no management changes are made (unfortunately, sometimes the case), a project will continue to run at that rate and will finish 15% behind schedule—hardly a good use of earned value analysis. Nevertheless, the linear extrapolations provide a starting point. To express these extrapolations compactly and consistently, we add another notational concept that elevates project discussions from a specific project to a general representation: projects begin at time zero and end at time one. From this normalization notion (as seen in Fig. 1), we use the symbol ‘‘1’’ to denote the end of a project. Furthermore, the prime mark ( 0 ) will denote quantities predicted from past performance. We begin with a prediction for the total project duration. The multiplicative schedule factor drives the new estimate of a projectÕs total duration straightforwardly DT 01
DT 1 F s ;

ð7Þ

where DT1 is the originally planned project duration and DT 01 is the new estimate. We can use a lower-case delta to express the difference between the new estimate and the originally scheduled duration:

dT 1 ¼ DT 01  DT 1 ¼ DT 1 ðF s  1Þ.

141

ð8aÞ ð8bÞ

What about costs? In this new notation, the final cost as originally planned (standard terminology: ‘‘budget at completion’’, or ‘‘BAC’’) is Cs,1, i.e., the scheduled cost at the end of a project. This number equals the total of all the previously budgeted individual costs. At the end of the project, the budgeted cost will often not equal the actual cost. In the standard language, earned value analysis is used to make an ‘‘estimate at completion’’, or ‘‘EAC’’ [3]. These initials contrast with the new notation, where we can write unambiguously the prediction ( 0 ) of a projectÕs actual cost (Ca) at its end (1): C 0a;1 . Historically, practitioners have used several different methods to predict this all-important number. I mention the first type of final cost prediction only to illustrate blind, unrealistic optimism. Fleming and Koppelman [3] note that some have called the ‘‘overrun to date’’ estimate ‘‘useless’’. It does not predict a future based on past performance so much as it ignores the past and pretends that a project will evolve according to its original plan. To obtain the final cost estimate with this method, one adds the current actual cost (Ca) to the cost of the remaining work, which is the difference between the planned total cost and the earned value (Cs,1  Cb), to obtain Cs,1  DCa. Another possible method, the ‘‘high-end cumulative CPI times SPI EAC’’, was designed to use the product of the traditional two indices as an inverse factor on the remaining work, but the new notation reveals it as a combination of schedule factors, actual costs, and the proper estimate: C a þ ðC s;1  C b ÞF s F a ¼ C 0a;1 F s C a ðF s  1Þ. The new notation shows that the final method listed by Fleming and Koppelman (‘‘low-end cumulative CPI EAC’’) is the straightforward analog of the duration prediction: C 0a;1
C s;1 F a .

ð9Þ

With this estimate finally in hand, we can write the difference between what has been spent (Ca) and what needs to be spent to complete the project. Improving upon an earlier version of this notation [4], we avoid confusion with the differences in Eq. (3) by using a lower-case delta: dC a
C 0a;1  C a ð10aÞ ¼ C a ðF s;1  1Þ;

ð10bÞ

where, consistent with the initial definition of the factors (Eq. 4a), Fs,1 ” Cs,1/Cb represents the total project value in terms of what has already been earned; the inverse of Fs,1 shows the fraction of the original budgeted cost that has been completed. When the actual cost equals the budgeted cost (i.e., when Ca = Cb), dCa reduces to Cs,1  Ca, as expected.

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D.F. Cioffi / International Journal of Project Management 24 (2006) 136–144

We can also write the difference between the new estimate of the final cost and the original estimate of the final cost: dC 1 ¼ C 0a;1  C s;1 ¼ C s;1 ðF a  1Þ.

ð11aÞ ð11bÞ

5.1. TCPI A final example of a translation from the lettered, traditional scheme again shows the utility of this new notation. Fleming and Koppelman [3] write that the ‘‘To Complete the Remaining Work Performance Index’’, or TCPI, ‘‘tells management what performance factor must be achieved on the remaining work in order to stay within the financial goals. . .’’ They present a figure and the following expression: Remaining Work ðBAC  EVÞ ¼ TCPI. Funds Remaining ð½BAC or EAC  ACÞ The text quoted above asks for a ‘‘performance factor’’, but the formula is termed an index. In the context of the standard notation, this mixed terminology presents no contradiction in principle because ‘‘factor’’ and ‘‘index’’ are not differentiated. One could therefore speak in a general way of an index, where a value greater than one shows performance better than planned, as a performance factor. However, this formulaÕs behavior depends on the choice made in the denominator. One choice gives a traditional index, and the other produces a factor as defined in this paper, where a number less than one indicates performance better than planned (Eq. (4)). Using the original estimate of the completed projectÕs cost (BAC) for the difference in the denominator produces a standard index. In the new notation, one finds C s;1  C b TCPI ¼ ð12aÞ C s;1  C a F s;1  1 . ð12bÞ ¼ F s;1  F a If a project is underperforming, Fa > 1. Therefore, the above expression must be greater than one because the numerator is greater than the denominator. The ratio does then represent a proper index, where a value greater than 1 asks for improved performance to meet the original plan. However, as in the earlier discussion of ‘‘overrun to date’’, using a revised estimate of the project cost (C 0a;1 ) makes more sense than using the estimate from the original plan. Choosing this ‘‘EAC-AC’’ instead of ‘‘BAC-AC’’ in the denominator now produces C s;1  C b TCPI ¼ 0 ð13aÞ C a;1  C a 1 ¼ . ð13bÞ Fa

When Fa > 1 (as before, underperformance), this ratio is less than one, i.e., the complete opposite of the prior result! To be interpreted as a call for greater effectiveness, the expression must be seen as a factor not in general but according to the definition used in this paper, where a number less than one means performance better than planned originally. Indeed, we might have guessed the answer immediately from the description of the quantity desired. We know that the cost factor gives the rate at which a project is falling behind in cost. In the zeroth-order view, then, work at the inverse rate will compensate for the underproduction. In this latter case, however, has this ratio been defined inconsistently? The numerator contains a remaining cost estimate based on the original plans, but the denominator calculates remaining funds based on the new estimate. More important, a project consists of many tasks, not one, and the neglect of some type of aggregate factor makes this simple view suspect from the start. 5.2. Earned value over multiple tasks Generally speaking, the accuracy of earned value analysis improves if one obtains the cost triplet from the progress of multiple project tasks rather than from a single task. Below we write the aggregate triplet costs from the sum of N individual tasks that were scheduled to occur up to some given time: Cb !

N X ðC b Þj

ð14aÞ

j¼1

Ci !

N X ðC i Þj ;

ð14bÞ

j¼1

where again i takes on a and s to produce Ca and Cs. We can write the aggregate factors, F
i , in terms of the above sums F
i


N N X X ðC i Þj  ðC b Þj . j¼1

ð15Þ

j¼1

If we can express the aggregate factor in terms of the individual factors, (Fi)j, ðF i Þj
ðC i Þj  ðC b Þj ;

ð16Þ

we will be able to see the importance of any one project task as compared to the others. We find this expression by defining a weighting factor, wj, that shows each taskÕs fraction of the total budgeted cost: wj
ðC b Þj 

N X ðC b Þk .

ð17Þ

k¼1

The expression confirms what we know intuitively: tasks with a greater cost budgeted can earn more value. The weighting factors themselves sum to 1:

D.F. Cioffi / International Journal of Project Management 24 (2006) 136–144 N X

wj ¼ 1;

j¼1

and they allow us to write the aggregate forms in terms of the individual factors F
i ¼

N X

wj ðF i Þj

ð18aÞ

j¼1

! wj ðF i Þj ;

ð18bÞ

where, if one wishes extreme compactness, the second view uses the summation convention, in which repeated indices (present on only one side of the equal sign) imply summation over the duplicated js. 6. Illustrations and conclusions In this final section, two illustrations show the potential power of this notation, and the conclusions include a reference to an improved treatment of the ‘‘to complete’’ problem. 6.1. Keeping cost constant The notation used to prepare for the earned value calculations, i.e., Eqs. (1) and (2), can demonstrate project management verbal truisms mathematically, which makes them explicit and enables more precise communication. For example, we know that if the duration of a task increases, the only way that the cost can remain constant is if we use fewer resources. Combining Eqs. (1) and (2) produces a single expression for task cost, C, in terms of task duration, Dt, resource intensity, RI, and cost rate, C_ R . The expression may be differentiated and set to zero to show the conditions to satisfy for constant cost C ¼ RI DtC_ R ð19aÞ ! dC ¼ 0 ¼ DtC_ R dRI þ RI C_ R dDt þ RI DtdC_ R .

ð19bÞ

Eq. (19b) shows that if the duration increases (i.e., dDt > 0) and the cost rate stays the same ðdC_ R ¼ 0Þ, the resource intensity must decrease for the cost to remain unchanged (dRI < 0 means fewer resources). On the one hand, we can consider this example a trivial one, but the compact notation provides the important information in one line. For example, the equation reminds us that contrary to the oversimplification used to introduce this illustration, we should not forget cost rates when examining resource use. 6.2. Increasing the importance of recent tasks The appropriateness of using earlier tasks to predict future performance may be questioned in either of two possible scenarios: the new tasks are mostly similar to

143

or different from the earlier tasks. If the tasks are similar, the project team may have increased capabilities because of its experience with the tasks. Hence, project managers may want to give greater weight to earned value predictions from later tasks. In the other scenario, performance on tasks at the beginning of a project may have little bearing on the different tasks occurring later as the nature of a project changes. Again, the project manager may wish to give lesser weight to the earlier tasks. With minor modifications, the above notation for earned value over multiple tasks can be used do generate solutions to this dilemma, and I present one here. We can give earned value quantities calculated from later tasks a greater weight simply by scaling them according to the time at which they occur. For example, tasks measured at a normalized time b = 0.75 (where the project ends at b = 1) could be worth three times as much in any earned value calculation as those measured at b = 0.25. However, because we typically calculate earned value quantities only at several times in a projectÕs life, we can use the proportionalities formed from those specific times for all tasks that finish within the intervals between adjacent measurement times, as shown immediately below. First, the time or schedule factors evaluated at a given normalized time, b(l), use only the N(l) 6 N tasks that have finished since the time of the previous earned value calculation, b(l  1). The factor weights, wj, are evaluated over the same subset of the total N tasks, and the index j begins anew within each interval b(l)  b(l  1): F
i ðlÞ


N ðlÞ X

wj ðF i Þj

ð20aÞ

j¼1

wj
ðC b Þj

X N ðlÞ ðC b Þk .

ð20bÞ

k¼1

The individual factors are weighted according to the time interval in which they were obtained to produce the projectÕs cumulative earned value factor: L X 1 bðlÞF
i ðlÞ; l¼1 bðlÞ l¼1

hF
i i
PL

ð21Þ

where the brackets, Æ æ, around the F
show that this average differs from that in which all project tasks are weighted equally. This particular configuration represents only one example of a non-equal weighting of earned value quantities across the life of a project. One can argue over the scheme (perhaps the weights are too severe), but the point is made: this notation facilitates a userÕs desire to develop a scheme more to the liking of the individual organization or to the individual project manager. Further research with actual project data may produce a favored mechanism with demonstrated accuracy.

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6.3. Conclusions This paper presents a new formalism that can solve the problem of the historically ‘‘arcane and ponderous’’[8] notation used in earned value analysis. I have reproduced the standard earned value parameters in this new formalism; Table 1 shows the translation of standard terminology into the new notation, and Table 2 shows additional quantities defined here. The symbols carry precise and distinct meanings. This modern symbolic notation for earned value calculations can accomplish several objectives: (1) Manipulation of the earned value parameters is easier and faster. (2) Project management will be elevated as a research discipline. (3) Because technical presentations about earned value would be easier to understand, project managers might use earned value analysis more frequently. (4) More use of earned value would further its development and lead to more streamlined processes and possibly to advanced and more useful methods (e.g., non-linear predictions). This notation should make smoother the path to more subtle and more productive uses of earned value. The two examples immediately above show how the notation can demonstrate common project management truisms mathematically (to aid instruction) and can facilitate new developments in earned value, such as giving higher weights to more recent tasks. Learning this formalism requires some study, but only elementary algebra lies behind the notation. The consistency and clarity that result show the advantages of this new view, and these benefits can assist the transition from earned value analysis to earned value management by a wider segment of the project management community. The notation has already been used on the problem of predicting future project efforts in terms of earlier work efficiency. An improved solution to this ‘‘to com-

plete’’ problem, which lies beyond the scope of this paper, is presented elsewhere [14]. Acknowledgements I thank Young Hoon Kwak for discussions that improved the presentation of this paper, and Eric A. Cioffi for assistance with the tables.

References [1] Kuhn TS. The structure of scientific revolutions. 2nd ed. The University of Chicago Press; 1970. [2] Simon HA. The sciences of the artificial. 3rd ed. Cambridge, MA: The MIT Press; 1996. [3] Fleming QW, Koppelman JM. Earned value project management. 2nd ed. Newtown Square, PA: Project Management Institute, Inc.; 2000. [4] Cioffi DF. Managing project integration. Virginia: Management Concepts, Inc.; 2002. [5] Rad PF, Cioffi DF. Work and resource breakdown structures for formalized bottom-up estimating. Cost Eng 2004;44(2):31–7. [6] Project Management Institute Standards Committee, A guide to the Project Management Body of Knowledge (PMBOK), 3rd ed. Newtown Square, PA: Project Management Institute, Inc.; 2004. [7] Frame D. The new project management. San Francisco: JosseyBass; 1994. [8] Rose KH. Review of earned value project management, 2nd ed. Project Manage J; 2003. [9] Sugrue CP, Corrado GS, Newsome WT. Matching behavior and the representation of value in the parietal cortex. Science 2004;304:1782. [10] Brown L, editor. The new shorter Oxford english dictionary. Oxford: Clarendon Press; 1993. [11] Cioffi DF. A tool for managing projects: an analytic parameterization of the S-curve. Int J Project Manage 2005;23:215–22. [12] Anbari F. An operating management control system for largescale projects. In: Paper presented at American Institute for Decision Sciences, ninth annual meeting, northeast regional conference, Philadelphia; 1980. [13] Anbari F. Earned value project management method and extensions. Project Manage J 2003;34(4). [14] Cioffi DF. Completing projects according to plans: an earned value improvement index, J Operat Res Soc in press [accepted 22 March 2005].