Project Management Institute

Designing project management

a scientific notation and an improved formalism for earned value calculations

Director, Project Management Program

The George Washington University

Proceedings of the PMI Research Conference 11-14 July 2004 – London, UK

1.0 Introduction

“If you can't measure it, you can't 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. No doubt this problem has existed since ancient Egypt, when the Pharaoh's project manager wondered how to estimate the remaining cost and duration of a nearly completed pyramid. At The George Washington University, we show our students the sketch below, which fortunately we were able to obtain from that project manager's journal. If the plan calls for a total of 1,000,000 stone blocks and 900,000 are now in place, is the project 90% complete?

Building the great pyramid at Gizeh

Exhibit 1: Building the great pyramid at Gizeh.

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 as a function of 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 U.S. Department of Defense released its first official list of “Cost/Schedule Control Systems Criteria” (C/SCSC) (Fleming and Koppelman, 2000), signaling the formal initiation of earned-value analysis, which still represents management’s best chance at measuring a project's 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 a roadblock to its embrace by the management community. In an attempt to evolve the system to a more scientific format, this paper puts forward a notation that I developed originally for myself and for my students, and that I first wrote about formally in Managing Project Integration (Cioffi, 2002).

1.1 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 project’s primary financial currency (e.g., dollars in the United States of America; pounds in Great Britain) 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, for example, 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. Before beginning the formalization of this transition, a brief explanation may be helpful for those not familiar with the use of Greek symbols in mathematical notation. The Greek letter delta (Δ and δ, uppercase and lowercase, respectively) is used traditionally to represent differences in quantities. A duration is considered the difference in time between one point and another, and so it is usually represented with Δt or some variation thereof. Thus, 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, Δt. In Managing Project Integration, I discuss the optimum duration of a task, where the duration and therefore the effort become functions of the resource intensity itself, but here we will write the common linear approximation:


To quantify the example offered above, let us use the resources attorneys and bulldozers in intensities of 1 and 10, respectively, for durations of 6 minutes and 2 days. The resulting efforts are 6 attorney minutes and 20 bulldozer days. One compares these efforts by converting them to costs through a cost rate, ĊR (read “C - R dot”), that might be obtained from a resource breakdown structure (Rad and Cioffi, 2002). 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,


where we have now properly introduced the generic cost, C, that will become specific in earned-value calculations.

1.2 Earned Value Defined

The earned-value system incorporates scope and integrates it with cost and schedule. First, the manager determines the value of a project’s 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. What a wonderful notion! Only when a specified amount of 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 shorthand we refer to as “earned value.”

In this framework, the mere contemplation of a 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, that is, according to a project’s schedule, at the specific time of any comparison.

1.3 Standard Notation

Prior to the latest edition of PMI’s A Guide to the Project Management Body of Knowledge (PMBOK® Guide — Project Management Institute Standards Committee, 2000), 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

Students (and practitioners) would grind through calculations with these sets of initials. Confusion resulted. As Frame put it ten years ago, “students spend more time trying to master the vocabulary than the concepts.” (Frame, 1994). My more recent observations as an instructor agree with this perspective. Moreover, in a recent review (Rose, 2003) of the second edition of Earned Value Project Management (Fleming and Koppelman, 2000), Kenneth Rose notes earned value’s “historically arcane and ponderous terminology.”

The Project Management Institute attempted to improve this terminology in the latest edition of the PMBOK® Guide by reducing 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. 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.

2.0 A New Formalism

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. A modern symbolic notation for earned-value calculations could accomplish several objectives:

  1. Manipulations of the earned-value parameters would be easier.
  2. Because presentations about earned value would be easier to understand, project managers would use earned-value analysis more.
  3. More use of earned value would further its development and lead at least to more streamlined processes and possibly to advanced and more useful processes (i.e., non-linear predictions).
  4. Project management would be elevated as a research discipline.

This use of “value” qualifies it as a “term of art” in the field of project management. 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 words’ 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 meanings to a new notation. In fact, one of the definitions of “formalism” refers to “manipulation according to certain rules of intrinsically meaningless symbols” (Brown, 1993), which explains why the terminology and the notation need not be identical: just as we had to define “value” as a term of art, we give meaning to symbols created to illuminate calculations instead of obfuscating them. At first, a user must put forth some additional effort to understand these new symbols, which will not replicate the definitions. However, the resulting ease in future manipulations should justify the initial outlay of effort.

What goals might we set for a new formalism?

  1. 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.
  2. It should be consistent. As much as possible, the same symbol should refer to the same quantity or operation whenever it is used.
  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, such as one cost divided by another.
  4. It should be relatively compact. Compact notation facilitates manipulation and understanding.

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. (Managing a project is, of course, all about managing changes as reality diverts from the plans.) A “scheduled” cost implies a “budgeted” cost, but it cannot be that of the work performed because one does not schedule costs that have already occurred, that is, actual ones. Below I show the evolution of the triplet’s notation:

  • Earned value:    BCWP → EV → Cb
  • Planned value:   BCWS → PV → Cs
  • Actual cost:      ACWP → AC → Ca

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 earned-value quantities almost instantly. With the subscripts, one would read “budgeted cost, “scheduled cost,” and “actual cost.” The novice would inquire about the difference between “budgeted” and “scheduled,” and he or she would be told that the earned-value system is designed to measure progress, and that 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.

Exhibit 2 illustrates these fundamental earned-value parameters on a cost versus time plot (an S curve) where the plotted values have been normalized to the project's total cost and total duration, respectively. (One transforms these dimensionless costs into costs with currency units by multiplying by the total budgeted cost.) In this example, the actual cost and the budgeted cost have been measured at 25 percent of the project's 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.

2.1 Earned Value In Multiple Tasks, Part 1

Generally speaking, the accuracy of an earned-value analysis improves if one estimates the cost triplets from the progress of multiple project tasks, rather than from that of a single one. We produce the aggregate triplet costs from the sum of the individual tasks, from 1 to N:


where, in an economy of notation, i takes on a and s to produce Ca and Cs. First, this compact notation saves space and time when writing and viewing these symbols. More important, it emphasizes their sibling nature and discriminates between them only when necessary.

3.0 A Scientific Notation

In this section, I combine the earned-value triplet parameters to show the standard derived quantities of earned-value analysis, and I shall also introduce a new one.

The Earned Value Triplet on a Normalized S-Curve

Exhibit 2: The Earned Value Triplet on a Normalized S-Curve

3.1 Differences

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 project varies from the 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. Again, we use the symbol that routinely represents differences, and with this new notation we can present both differences with a single equation:


where “i” becomes “a” to produce the cost difference and “s” to produce the schedule difference. When the quantity ΔCi is negative, a project is over budget (i = a) or behind schedule (i = s). Exhibit 2 indicates how one would measure these differences on a plot that shows an S curve of normalized cost versus normalized time.

3.2 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. However, inverting the indices, suggested at least once in the literature (Anbari, 1980), 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 (which will be examined shortly) that examines both cost and schedule together.
  2. The ratio shows immediately the fractional differences between planned and actual costs and schedule.

Again, a single equation defines both factors:


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 Fi = 1 identically. Unlike the standard indices, a project is under budget or ahead of 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


percent over budget or behind schedule. (If Fi < 1 a project is under budget or ahead of schedule, and the above difference is negative.) This expression can also be written as


The ratio ΔCi ÷ 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 §4.

3.3 Earned Value In Multiple Tasks, Part 2

Aggregate factors images (read “bar F sub i”) can be expressed in terms of the sums defined previously (Equation 3),


If we can express the aggregate factor in terms of the individual factors, (Fi)j,


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 task’s fraction of the total budgeted cost:


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


and they allow us to write the aggregate forms in terms of the individual factors:


In the expressions to come, the factors are written without a bar above them, but an analysis of a real project should use the aggregate factors. The lack of the bar does not affect the algebra that follows in this paper.

3.4 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 numbers. The minimum number would be one indicator per project, and I suggest the possibility of the following combination for the cost-schedule performance factor:


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 Φ = 1.5, which highlights quickly the problem waiting to be discovered in the project's schedule (the discrepancy between the factors may also be problematic).

Some earned-value users maintain that the product of the standard cost and schedule indices, called the “Critical Ratio” (Anbari, 2003), 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 that everything is fine, but most executives would want to know if one of their projects was 33% behind schedule (Fs – 1 = 0.33), and Φ = 1.5 would alert them quickly when glancing down a single column of numbers.

4.0 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 factors above (initially defined in Equations 5) describe not just the past of a project, but its future too. If earned-value measurements show that a project is 15% behind schedule (Equation 6a: 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 idea, we use the symbol “1” to denote the end of a project (as used in Exhibit 2). Furthermore, the prime mark (׳) will denote quantities predicted from past trends. We begin with a prediction for the total project duration.

The schedule factor drives the new estimate of a project’s total duration, ΔT1. It is given straightforwardly:


We can use a lower-case delta to express the difference between the new estimate and the originally scheduled duration:


What about costs? In this new notation, the final cost as originally planned (“budget at completion,” or “BAC”) is Cs,1, that is, the scheduled cost at the end of a project. This number must equal the total of the budgeted costs, and so it can be used for the sum in Equation 9:


At the end of the project, the scheduled cost will probably not equal the actual cost. In the standard language, earned-value analysis is used to make an “estimate at completion,” or “EAC” (Fleming and Koppelman, 2000). These initials contrast with the new notation, where we can write unambiguously the prediction (′) of a project’s actual cost (Ca) at its end (1): C′a,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 themselves note that some have called the “overrun to date” estimate “useless.” It does not predict a future based on past performance as 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 remaining work, that is, the difference between the planned total cost and the earned value (Cs,1Cb), to obtain Cs,1 – ΔCa.

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: Ca + (Cs,1Cb) Fs Fa = C'a,1 FsCa (Fs 1).

The new notation shows Fleming and Koppelman's last listed method (“low-end cumulative CPI EAC”) to be the straightforward analog of the time prediction:


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 the notation used in Managing Project Integration (Cioffi, 2002), we avoid confusion with the differences in Equation 4 by here using a lower-case delta:


where, consistent with the initial definition of the factors (Equation 5a),


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), δCa reduces to Cs,1Ca, as expected.

We can also write the difference between the new estimate and the original estimate of the final cost:


4.1 Designs and Revolutions

Thomas Kuhn introduced the oft-misused term “paradigm shift” in his seminal book, The Structure of Scientific Revolutions (Kuhn, 1970). Scientists build models or “paradigms” to explain the universe's physical and biological structures and activities. 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 show 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, for example, special relativity replaces Newtonian dynamics at speeds near that of light, and quantum mechanics must be used when investigating phenomena at atomic and sub-atomic scales. Such new models give a new perspective that provides a better explanation of the world in which we live.

In a manner analogous to shifts in scientific theories, earned-value analysis started 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. Management should no longer accept separate views of budget or schedule. 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 (Simon, 1996), which comprise project management.

According to Simon (page 111), “Everyone designs who devises courses of actions aimed at changing existing situations into preferred ones…Design, so construed, is the core of all professional training….” 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 (hoped-for?) paradigm shift.

5.0 Conclusion

I have reproduced the standard earned-value parameters in a new formalism. The symbols and the associated sub-symbols in the resulting notation carry precise and distinct meanings.

Learning this formalism requires study, but the consistency and clarity that result will be worth the effort. These benefits should assist the transition from earned-value analysis to earned-value management by a wider segment of the project management community. Former students have told me that technically trained people like the notation. However, behind the symbolism is only elementary algebra that an educated person can handle with a small investment of time.

Finally, I hope that this notation makes smoother the path to a non-linear version of earned value.

6.0 Acknowledgments

I thank the Management Science Department at The George Washington University for the laptop computer that enabled me to sit quietly, away from e-mail and the Internet, and complete this paper. I also thank Dr. Bill Wells for the peek into the journal of the Pharaoh's project manager.

7.0 References

Anbari, F. (1980). An operating management control system for large-scale projects. In American Institute for Decision Sciences, Ninth Annual Meeting, Northeast Regional Conference, Philadelphia.

Anbari, F. (2003). Earned value project management method and extensions. Project Management Journal, 34(4).

Brown, L., editor (1993). The New Shorter Oxford English Dictionary. Clarendon Press.

Cioffi, D. F. (2002). Managing Project Integration. Vienna, VA: Management Concepts, Inc.

Fleming, Q. W., & Koppelman, J. M. (2000). Earned Value Project Management, second edition. Newtown Square, PA: Project Management Institute.

Frame, D. (1994). The New Project Management. San Francisco: Jossey-Bass.

Kuhn, T. S. (1970). The Structure of Scientific Revolutions, second edition. The University of Chicago Press.

Project Management Institute Standards Committee (2000). A Guide to the Project Management Body of Knowledge — 2000 edition. (PMBOK® Guide). Newtown Square, PA: Project Management Institute.

Rad, P. F. & Cioffi, D. F. (2002). Resource Breakdown Structure (Chapter 3). In Project Estimating and Cost Management. Vienna, VA: Management Concepts, Inc.

Rose, K. H. (2003). Review of Earned Value Project Management, second edition. Project Management Journal (September).

Simon, H. A. (1996). The Sciences of the Artificial, third edition. The MIT Press.

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