# A macro earned value model

**Department of Administrative Sciences**

**Metropolitan College, Boston University**

**Introduction**

Earned value management (EVM) is a methodology that attempts to determine the actual progress of a project, as opposed to the proposed or planned progress. Several useful introductions to the EVM approach exist (such as Anbari, 2003; Project Management Institute, 2004; Vandevoorde & Vanhoucke, 2006). EVM has also been covered in many textbooks (such as Kerzner, 2005). Christensen (2007) provided a comprehensive, research-oriented bibliography on the EV topic. EVM has also been widely used to analyze and monitor the status of real-world projects (for example, Christensen, 1998).

According to the Project Management Institute's *Guide to the Project Management Body of Knowledge (PMBOK ^{®} Guide*) (2004), earned value (EV) is the value of work performed expressed in terms of the approved budget assigned to that work for a schedule activity or work breakdown structure (WBS) component. EV provides visibility into costs and schedules by providing a measure of the difference between planned costs and actual costs spent.

Flemming and Koppelman (2000) provided a history of EV from its origins in the Department of Defense to eventually becoming its own methodology in 1967. Much of the literature on EVM can be traced back to the work of Flemming and Koppelman. Sorel (2004) said the Federal Highway Administration has used the EV method effectively on large projects. However, EVM is not without its critics; for example, Pailen (2007) suggested that EVM is complicated and difficult to implement, and as a result many EVM implementations have been unsuccessful.

Evaluation of the cost status of a project is measured in terms of two indices: the cost variance (CV) and the cost performance index (CPI). CV is the difference between the earned value (EV) and the actual cost (AC), (CV = EV − AC), where EV measures the work done (that is, earned) while AC measures the money actually spent. CPI is the ratio of EV to AC. To evaluate the progress of the project with regard to its schedule, EVM uses two more indices: the schedule variance (SV) and the schedule performance index (SPI). SV is the difference between the earned value (EV) and the planned value (PV), which is the work that should have been completed to date (SV = EV − PV). SPI is the ratio of EV to PV.

When CV > 0 and CPI > 1, the project is ahead of its cost budget. When SV > 0 and SPI > 1, a higher volume of work has been earned (that is, completed) than planned, and the project is ahead of schedule. However, SV, which is supposedly a measure of the schedule variance, is expressed as a monetary unit. This has led to criticism of the earned value schedule indicators by Lipke (2003) and Vandevoorde and Vanhoucke (2006). Henderson (2003, 2004) suggested that measuring SV and SPI in monetary units (and not in time units) makes them difficult to understand and is a source of confusion. Also, if SV = 0 (or SPI = 1), this could mean that either an activity is completed, or that the activity is running according to plan.

Also, toward the end of the project, all of the activities are nearing completion and so SV → 0, even if the project is late (similarly, SPI → 1). It has been suggested, therefore, that at a certain point in time the schedule indicators become unreliable, and this is generally considered to be two-thirds of the way through the project (Lipke, 2003; Vandevoorde & Vanhoucke, 2006).

To address these issues, several authors have proposed a concept called earned schedule (Corovic, 2007; Lipke, 2003, 2004; Lipke & Henderson, 2006). The earned value is traced forward or backward to the performance baseline—the planned value, PV. This determines an intersection point on the time axis and generates a “time-based” earned schedule (ES), which makes it easier to interpret. Henderson (2005) recently demonstrated the validity of ES by applying it on a portfolio of six projects.

While macro cost models exist for many industries, it appears that macro Earned Value models do not. I attempt to remedy this shortcoming by presenting an approach to macro EV. The model is illustrated by application in two industries where labor curves are well established: software development and construction. The results and conclusions of the two analyses are quite similar, and so it appears that the macro EV model may have wide applicability.

Using well-established labor curves from two completely different industries, I calculate the EVM indices over time. This allows me to predict the performance of both the cost and schedule indexes as a function of time, and in turn, to explain in detail their behavior. Some of the criticisms of the EV method can be addressed, because understanding of their behavior is improved. At the end of the paper, I explore the project management implications and suggest some options for future research.

Macro software project cost estimation models, and in particular labor curves, were pioneered by Putnam (1978), who used the labor curve previously suggested by Norden (1958). The model offers a fairly reliable method for predicting project completion times and labor requirements over time. Pillai and Sukumaran Nair (1997) claimed the model was unreliable during the initial phases of the project, but it seems to work well for complex systems, such as those including embedded hardware and software (Warburton, 1983). Putnam's work was later followed by Boehm's classic COCOCMO II model (1982) which has much wider applicability. The value of Putnam's work in this context is that it provides an analytical formula for the labor curve.

For construction projects, Wideman (1994, 2001) provided excellent examples of labor curves. Wideman also presented some rules of thumb that calibrate construction labor curves, which are based on previous work by Allen (1979). The general labor profiles appear to be quite typical (Christian & Kallouris, 1991), whether applied to a whole construction project, a sub-contract, or a continuous construction activity of significant duration. When plotted cumulatively over time, the typical labor profile seems to result in the typical S curve.

The ability to make schedule forecasts without performing a complete bottom-up schedule analysis of the work remaining has been long desired by EVM practitioners (Corovic, 2007). The model presented here may help with that goal by providing project managers the ability to develop early macro estimates of the project's EV performance over time, and then allow validation and cross-checking of the EV's actual progress as the project evolves.

**Project Labor Curves**

Putnam (1978) pioneered the use of the Norden-Rayleigh curve to describe the number of people working on complex software projects. The Putnam-Norden-Rayleigh curve, now known as PNR, appears to apply to many types of software projects, particularly embedded software systems (Warburton, 1983). The PNR labor curve is shown in Figure 1. The number of people working on a project as a function of time, *m(t)*, is given by:

T is a constant that denotes the time at which the number of people is at the maximum—the labor peak. K is a constant that can be determined by the condition that the total cost of the project—the total number of man-years—is *N*, where:

It is worth emphasizing that the PNR curve is the instantaneous labor, not the cumulative labor. Figure 1a plots this instantaneous labor (Equation 1) for the case where the total number of activities is *N = 100*, and the time of the labor peak is, *T = 10.* Figure 1b plots the cumulative version of the labor for the same data.

**Figure 1. The Putnam-Rayleigh-Norden (PNR) Curve Showing the Number of People Working on a Project as a Function of Time**

**Planned Value**

The rate at which activities are planned to be completed is defined as the planned value (PV). As labor is assigned, activities are completed. Therefore, it follows that the PV rate follows the same curve as the instantaneous labor rate:

*PV _{I}* denotes the instantaneous planned value of activities to be completed at time

*t.*As activities are staffed, there is no guarantee that they are completed on time, and so Equation 3 represents the planned completion of activities. Traditionally, PV is defined as the cumulative sum of the previous, instantaneous

*PV*(t), and so is defined as:

_{I}The instantaneous and cumulative planned values are plotted in Figure 2 (the dotted lines). The traditional cumulative version (Figure 2b) shows the typical S curve associated with cumulative costs. According to investigations by Singh and Lakanathan (1992), the application of S curves for cash flow projections can achieve accuracies of approximately 88–97%, and the shape of the S-curve budget versus time is a quick way to judge performance. This is precisely the objective of macro estimation methods: an early estimation of project performance from overall system parameters.

**Figure 2. The Actual Cost (solid line) and Planned Value (dotted line) as a Function of Time**

As the project proceeds, not all activities will be completed on time and so I now assume that a constant fraction, *α*, of the activities that are supposed to be completed at time *t* are rejected for some reason and require extra work. In the examples that follow, I use the value *α* = 0.2. While this number may seem high, it is useful to exaggerate it somewhat to clarify the behavior in the figures. Project managers should be able to estimate this parameter early on in the project as the initial activities are completed. Software project data suggests that error rates remain constant over the life of a project, so the availability of an early estimate of *α* is a reasonable assumption.

The completion of rejected activities will be delayed, and I assume that this delay is a constant amount, *τ*. That is, each activity that is rejected is delayed the same amount, *τ*. Unlike the error rate (or the rejection rate), there is little data on the average delay experienced by individual activities. Therefore, the reasonableness of this assumption needs to be evaluated by comparing real project data with macro models such as this one. In the examples that follow, I use the value *τ* = 3, representing an average delay of 3 weeks for each activity.

An example of the process is as follows: Suppose that at week 2 (t = *2)*, there are five activities that are due to be completed. One of these activities (α = 20%) is rejected for some reason. This one activity is delayed for rework and takes an extra three weeks to be completed, and so is completed in week 5 (*τ =* 3).

More formally, at time *t + τ*, the amount of extra work is simply *α m(t)*. Thus, *α* actually represents a combined average of several parameters, including the reject rate and the amount of rework required. To keep the model easily accessible, I have combined all of these effects into a single parameter.

**Defining the End of the Project**

Figures 1a and 2a show that the labor curves have long tails, which only slowly approach zero. Therefore, I will define the “end” of the project as the point in time when the cumulative curve shows 99.95% of the work is complete (that is, only 0.5% of the work remains). I will apply this definition to both the planned and actual curves.

**Actual Costs**

I can now calculate the instantaneous Actual Cost, *AC _{I}* (t), as follows: for

*t < τ*, the instantaneous cost is the same as the labor curve. The cost includes the work performed on both the activities that were successfully completed and those that were rejected. The extra costs for the rejected activities will be accounted for when the activities are completed at

*t +τ*. Therefore, in this stage the actual cost is simply the cost of the entire labor at this time; that is, for

*t < τ*:

For *t > τ*, the instantaneous labor cost is again m(t). This cost again includes the work performed on both the activities that were successfully completed and those that were rejected. The rejected activities will be completed at time *t +τ*. At time *t*, however, the activities that were rejected at time *t −τ* will now be successfully completed. Therefore, the instantaneous actual cost at time *t* is:

Equation 6 says that the actual cost is the labor expended at time *t* and an additional amount to correct the activities that were rejected at time *t − τ*. I can now calculate the cumulative actual cost, *AC*(*t*). However, the integration must be computed separately in the two time intervals:

*a)* 0 < *t ≤ τ*

In this interval, the cost in each week is simply the labor. Some of the activities will be rejected, but the costs for these will be accounted for when the activities are finally completed at *t + τ*. Therefore, the actual cost is simply the sum of the instantaneous labor:

*b) t >τ*

In this region, the cost in each week is again the labor. However, the costs of the rejected activities from the previous time, *t − τ*, must be added. Therefore, the cumulative actual cost as a function of time is:

The cumulative actual cost as a function of time is shown in Figure 2 (solid line), along with the planned cost (dotted line). The actual and planned costs are identical up to time *t = τ*. After that, the actual cost increases as the cost of the rejected activities accumulates. The actual cost as *t* → ∞ m represents the total cost of the project, which from Equation 8 is:

Thus, when a fixed percentage of activities are rejected for quality reasons, the result is a cost overrun for the project by the same percentage. It is interesting to note that in this model the total project cost does not depend on the time delay, *τ*, but only on the fraction of activities that were rejected, *α*.

**Earned Value**

The earned value is the value, or cost, of the successfully completed activities. In each interval, a fraction, *α*, of the activities were rejected, so the remaining activities, (1 *−α*), were completed and have earned value. Separating the time intervals, as before, the instantaneous earned value is:

a) 0 < *t ≤ τ*

In this region, the Earned Value is simply the fraction of activities successfully completed:

b) *t > τ*

For *t > τ*, the earned value has two components: 1) the fraction of activities successfully completed at the current time; and 2) those rejected at time *t−τ* that are now complete. Therefore, the instantaneous earned value is:

**Figure 3. The Instantaneous Earned Value (solid line) and the Instantaneous Planned Value (dotted line) as a Function of Time**

Figure 3a shows the instantaneous earned value (solid line) as a function of time as compared to the planned value (dotted line). The EV is initially delayed relative to the planned value, but eventually catches up. It is not evident in the curve because the value at the tail is so small, but the earned value is in fact slightly delayed at the end.

The cumulative earned value (EV) is found by the same process as for the cumulative actual cost: integrating in the two time intervals. The result is:

Figure 3b shows the cumulative earned value as a function of time (solid line) as compared to the cumulative planned value (dotted line). Both curves approach the same value as *t* → ∞ because the total number of activities in the project remains the same, *N (N=100).* The earned value is delayed because some of the activities were rejected and credit for their work was only earned when they were finally completed after the delay time.

If extra activities had been added (scope creep occurred), then the earned value would approach a higher level than the planned value. Since the number of activities has not changed, the total earned value must be the same as the total planned value. This is confirmed by Equation 13, which shows the Planned and Earned Values as *t* →∞:

**Cost and Schedule Performance Indexes**

The cost performance index (CPI) is defined as the ratio of earned value to actual cost, while the schedule performance index (SPI) is defined as the ratio of cumulative earned value to cumulative planned value (PMI, 2004):

Both CPI and SPI are traditionally defined in terms of the cumulative values. One advantage of this model is that one can calculate both instantaneous and cumulative versions of CPI and SPI. However, I will follow tradition and compute the cumulative versions, which are plotted in Figure 7.

**Figure 7. Cumulative CPI (bottom curve) and SPI (top curve) as Functions of Time**

The performance over time of the two indexes is interesting. CPI immediately falls to 1 − *α* because in each interval, including the first, some activities are rejected. As the rejected activities are completed after time *t = τ*, credit is earned, and the CPI rises. From Equation 14, the final values of CPI and SPI can be calculated:

At the end of the project, the CPI does not approach 1, it approaches the value shown in Equation 15, which depends directly on the reject rate. However, the behavior of the SPI is quite different. It also falls immediately to 1 −*α*. The SPI eventually climbs back to 1 at the end of the project, as it should. Table 1 lists the starting and ending values for CPI and SPI.

**Table 1. Starting and ending values for CPI and SPI.**

t → 0 | t → ∞ | |

CPI | 1 −α | |

SPI | 1 −α | 1 |

It is constructive to discuss the behavior of the SPI over time. A criticism of the use of SPI is that since it approaches 1, it is not useful over the last third of the project (Corovic, 2007; Fleming & Koppelman, 2003; Lipke, 2003, 2004; Henderson 2004; Vandevoorde & Vanhoucke, 2006). However, Figure 7 shows precisely how the SPI approaches 1. Knowing the precise time-dependent behavior of the SPI somewhat blunts this aspect of the criticism because with this model one can compare the SPI's actual performance to the projected performance.

The criticism that the SPI is in monetary units is still valid. However, the earned schedule process can be applied just as well in this model. In fact, using this model, one can develop the macro estimate the earned schedule over time, and then compare the actual earned schedule as it evolves over the life of the project. This is a topic for further research.

**Construction Labor Curves**

I now turn to analyzing labor curves that are typically applied in the construction industry.

**Allen's Labor Loading**

Wideman (1994) provided labor curves for a profitable civil contract that was predominantly formwork and concrete placing. The data is redrawn in Figure 8a, which shows a histogram of the production workforce over the 38-week project duration. There is an initial period of build-up, a period of peak loading, followed by a period of progressive demobilizing. This instantaneous profile, when plotted cumulatively over time for the whole project, results in an S curve. Wideman presented some rules of thumb relating to these curves, which appear to be quite typical (Christian & Kallouris, 1991), whether the observations refer to a whole construction project, a sub-contract, an individual trade, or a continuous construction activity of significant duration.

**Figure 8a. Actual (histogram) and Approximate (trapezoidal) Labor Curves for a Construction Project**

**Figure 8b. Comparison of Construction (trapezoidal) and PNR Labor Curves**

Allen (1979) suggested that a simple trapezoidal figure can be used as a good approximation of the actual labor loading. The trapezoidal profile shown in Figure 8a consists of a linear ramp-up to a peak after 40% of the schedule, a constant peak load ending after 75% of the schedule, and a ramp-down to the end. In Figure 8a the agreement between the actual project data and the approximate loading is quite striking.

For comparison, the PNR and trapezoidal curves are shown on the same scale in Figure 8b. The PNR curve was scaled so that it has the same total labor as the construction curve, so the two are directly comparable. The PNR curve has a faster build up and a longer tail, but a peak of roughly the same height.

I shall denote the trapezoidal curve as *c(t).* The peak value of the labor curve is defined to be P, and the project ends at time *T _{e}*. The ramp up spans the interval [0,

*T*], and using Allen's Rule #4 (the peak occurs after 50% of the project),

_{a}*T*/2. The constant section spans the interval [

_{a}= T_{e}*T*,

_{a}*T*], and using Allen's rule #5 (the peak occurs from 50% to 75% of the project), the value of

_{b}*T*/ 4. Thus, the equation for the trapezoidal curve is:

_{b}= 3T_{e}As in the PNR labor curve case, the planned value (PV) for the trapezoidal curve is simply the cumulative sum of the history of the instantaneous values:

The integrals in Equation 17 are quite standard. An estimate of the total labor can be obtained by evaluating the second Equation 17, and substituting the values from Figure 8a for the peak *P* and the time *T _{e}*. One obtains a prediction for the total labor of the entire project of 2,048 labor weeks. This is an excellent macro approximation to the actual value of 1,994 labor weeks given by Wideman (1994). The error is only 2%.

As a further demonstration of the power of the macro analytical model, I can also derive Allen's Rule #1: the peak labor, P, is 1.6 x average labor. The total labor is given by Equation 17, so the average labor is obtained by dividing the total labor by the total time, *T _{e}*, which gives:

This is Allen's Rule #1, and demonstrates that there is actually redundancy between the rules. However, Equation 18 also demonstrates that the proposed macro EV model agrees with empirical results previously well established.

**PV, AC, and EV for the Trapezoidal Curve**

The analytical process previously followed for the PNR curve can now be repeated for the trapezoidal labor curve, Equation 16. The integrals are somewhat tedious, but again, quite straightforward. Figures 9 through 11 summarize the results.

Figure 9a presents the instantaneous planned value (dotted line) and the instantaneous Actual Cost (solid line) for the trapezoidal labor curves. At the end of the project, *t = T _{e}*, there is a small amount of extra work that is clearly visible. This represents the work rejected in the interval

*t = T*that is completed after the planned end of the project,

_{e}− τ*t = T*. Figure 9b presents the cumulative version of the same data.

_{e}**Figure 9. Planned Value (dotted line) and Actual Cost (solid line) for Construction (trapezoidal) Labor Curve**

Figure 10a presents the instantaneous planned value (dotted line) and the instantaneous earned value (solid line) for the trapezoidal labor curves. Figure 10b presents the cumulative version of the same data. The planned value and earned value curves are hard to distinguish, especially in the cumulative version. The curves are also very close to the PNR curve, Figure 3. One of the recommendations of this model, therefore, is that project managers should be encouraged to use instantaneous labor curves in which the deviations of earned values (EV) from planned values (PV) are more evident. Using the instantaneous versions, project managers should be able to better interpret any deviations and take corrective action.

**Figure 10. Planned Value (dotted line) and Earned Value (solid line) for Construction Trapezoidal Labor Curves**

**CPI and SPI for the Trapezoidal Curve**

The CPI and SPI for the trapezoidal construction curve are shown in Figure 11a. Figure 11a shows that the CPI immediately falls and remains low over the entire life of the project. In comparison, the SPI falls initially, but then slowly climbs back to 1, as expected. In Figure 11b, the CPI and SPI curves are repeated for the PNR labor curve, so that they can be easily compared.

**Figure 11. CPI and SPI for PNR and Construction (trapezoidal) Labor Curves as a Function of Time**

Despite the quite different labor curves (see Figure 8b), the resulting behavior of the CPI and SPI over time is remarkably similar in the two cases. Christian and Kallouris (1991) established that for most projects the typical cumulative labor profile seems to result in an S curve. This means that for most projects, the instantaneous labor will be quite similar to those of Figure 8b.

This suggests an intriguing and potentially significant property of this macro EV model: the behavior of the CPI and SPI curves over time for most projects should be similar to those shown in Figure 11. Therefore, as long as the labor curves follow the general shape in Figure 8b, which will in turn result in the usual S-shaped labor curve, the tentative conclusion arrived at here is that the resulting CPI and SPI curves should resemble Figure 11. That is, CPI and SPI over time will be relatively independent of the precise form of labor curve. This suggests that the model has wide applicability, and might be quite general. This is not totally surprising, because macro software cost models have been shown to have a surprisingly wide applicability. Since the PNR labor curve is also the basis of this macro EV model, it should inherit its general features and applicability.

**Conclusions**

I have proposed a simple macro earned value model with some powerful predictions. The model was applied in two cases, the PNR labor model for software projects and Allen's trapezoidal labor curve for construction projects. What emerged was the behavior of the CPI and SPI curves as a function of time.

The macro EV model requires two constants: the rejection rate and the time delay to repair the rejected activities. Since not all activities will be completed on time, I assumed that a constant fraction of them were rejected and required extra work. Project managers should be able to estimate the required parameters early on in the project as the first few activities are completed. I further assumed that the rejected activities were all delayed by the same constant amount. Unlike the rejection (or error) rate, there is little data on the average delay experienced by individual activities. Therefore, it is a topic for future research to evaluate the reasonableness of this assumption.

Despite the apparent dissimilarity of the PNR and trapezoidal labor curves, coming as they do from quite different industries, they produce very similar behaviors for the CPI and SPI over time. The new information in the macro EV model presented here is that the specific performance of the SPI over time is now available and understandable.

Figure 11 also directly addresses the criticism that the SPI is not useful because SPI→1 (Corovic, 2007). However, this presupposes that the form of the SPI over time is not known. Once the precise form of the SPI over time is known, as is shown here, then the actual, observed performance can be compared to the planned behavior. The model explicitly calculates the SPI behavior over time, and the property that SPI→1 is, in fact, reasonable and expected.

This macro EV model is a preliminary effort. However, it has been reasonably established that the behavior of the CPI and SPI as a function of time can be calculated, and for a variety of industries with different labor curves. The behavior of the CPI and SPI curves over time for both labor shapes was similar, leading to the tentative conclusion that the CPI and SPI over time are relatively independent of the precise form of labor curve. This was based on Christian and Kallouris's (1991) observation that most project curves follow S shapes and so have quite similar instantaneous labor profiles. This suggests that the model has wide applicability. Since Allen and PNR labor profiles were used as the foundation of this macro EV model, it should inherit the general features and wide applicability of such models.

More complex issues need to be included and are topics for future research. Scope creep increases the number of tasks and so increases both actual cost and the earned value, which affects both the CPI and SPI. Also, it was assumed that activities were independent, which is clearly not true, as the critical path depends on the connection between activities. This might be addressed by assuming that the delay in one activity results in the delay of other activities further down the path. This should magnify the effect of the schedule delay and will presumably make the model more sensitive to the schedule delay parameter.

A macro EV model is a missing feature from the project manager's toolbox. Despite the simplicity of the macro EV model presented here, a number of interesting features emerged. While much work remains to be done, it appears that the model is a useful starting point and shows some promise.

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© 2008 Project Management Institute

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