Abstract
A critical examination of the approximations involved in the standard execution of earned schedule is presented first; then, a formal analytical definition is provided, eliminating ambiguities and clarifying the implications of the method's use. Once a typical S curve is assumed, the formalism requires no approximations, and thus the earned schedule concept is put on a sound theoretical basis. Exact analytical expressions are derived, and the simple result appears to have valuable, practical consequences: earned schedule is directly proportional to time. Furthermore, at early (project) times, the critical parameter can be determined directly from earned value's standard schedule performance index. Under the assumption of constant scope, this parameter does not change with time, and so once it is found, the system's evolution is fixed: the future is predictable.
Keywords: earned value management; earned schedule; schedule estimation; cost estimation
Introduction
From its first days as a monitoring system, the strength of earned value—its beautifully integrated treatment of schedule and budget—has confused many of those attempting to interpret earned value's non-intuitive measurements of schedule progress. Beginning with the definition of earned value itself, as the budgeted cost of work performed, the system first translates the planned schedule into planned budget expenditures and then it reports any delay in schedule as a difference between the planned spending and the earned value (in the fanciful official parlance, the “schedule variance”). Project managers schooled in the system soon learn to view the schedule in terms of currency, but most clients (and many hierarchical superiors) require additional explanation if told, “the project is US$10,000 behind schedule.”
The earned value measurement system also suffers from the coincidence of the earned value and the planned value at the end of any successfully completed project. Because one sums the earned values and the planned values from individual tasks, by the end of a (successful) project, all value that has been planned (i.e., the entire project), has been earned: any prior difference between planned value and earned value has vanished, and the “schedule variance” is zero, no matter how far beyond the planned completion date one might be.
Thus, as might have been expected, efforts arose immediately to translate any currency difference back into a temporal difference with, perhaps, the best candidate now being the so-called earned schedule method, which has had some success (Vanhoucke & Vandevoorde, 2007). In this paper, we challenge the original representation of earned schedule, examining some of its problems, but we also present a more formal definition of the concept and introduce some clarifying notation to remove ambiguity. Finally, we will show how our earned schedule equivalence has the potential of a direct, simple application to estimating project delays under the standard assumption that the project evolves in an S-curve fashion (e.g., Cioffi, 2005).
A Critical Evaluation of Earned Schedule
The “Schedule Variance Method,” which is based on earned schedule, outperformed other forecasting methods (e.g., the Cost-to-Complete), in forecasting total project duration (Vanhoucke & Vandevoorde, 2007). This improvement was not altogether surprising, because this method employs an instantaneous metric, continually re-estimating the change in schedule based on project data, compared with the Cost-to-Complete method, which is defined in terms of the global quantities assumed to be constant (e.g., budget at completion).
But what exactly is earned schedule? Lipke (2003) showed a phenomenological construction procedure for earned schedule and later explained it further in his book (Lipke, 2010). We investigate its fundamental structure more closely and explain some of the problematic issues that arise because of the current definition.
Figure 1: Planned Value, Cs, and Earned Value, Cb, for an S-shaped labor profile. The amount of schedule “earned” is denoted by the arrow from Cb (t) to Cb(t – δ).
First: A Note About Activity Correspondence
When the horizontal line is drawn from the earned value curve to the planned value curve to show the earned schedule, it introduces a potential mismatch in the activities, as illustrated in Figure 1 by the zones labeled A and B. The planned value, of course, represents the activities that are planned to be completed. During project execution some of these activities will be completed, others will not. The shaded area in the figure denotes completed activities. Section A represents activities completed that were not planned to have been completed. Section B represents activities not completed that were planned to have been completed.
By setting the earned value equal to the planned value, we are asserting a direct correspondence between the activities. More specifically, we assert that the activities completed in Section A have a cost equal to those not completed in Section B. If one assumes that the activities are similar and that average performance measures apply across the project, then the proposed correspondence between the two sections is reasonable. In this sense, therefore, ES is an average measure.
The Standard Expression for Earned Schedule
The equations that define earned schedule are written in terms of the more common earned value parameters
where i is an integer number of time periods (e.g., days, weeks, months, or years). The subsequent dimension-less ratio, Δ, presents an algebraic difficulty (Book, 2006) that explains itself if the denominator of the ratio, the PV difference, is instead understood as the slope of the planned value curve between i and i + 1:
The standard ES practice of using i as a subscript is not at all standard because it is a parameter, and traditionally, subscripts are used as identifiers, not parameters. Because not all these i parameters are the same, this confusion may have helped lead to some of the problems with ES presented here.
We can re-write the ES expression in terms of the slope, m, and the usual earned value schedule variance evaluated at time i:
where SV(i) ≡ EV(i) – PV(i). This expression shows that despite what has been written about ES being a new approach, it is based not just on the standard earned value parameters (as is acknowledged) but also on one of the standard derived parameters, the schedule variance, SV. The parameters that follow have a similar dependence. More important, this expression also shows that at a given schedule variance (i.e., earned value's measure of the schedule delay) taken at any particular project time, i, the derived earned schedule number, depends on the slope of the planned value curve, not just on the performance of the project.
First, this new way of presenting the ES showcases an important conceptual problem. Why should the performance of the system depend on the local planned value cost rate at about the time of the measurement? Arguments could presumably be made for a dependence on some thoughtfully chosen average over some combination of previous points (not the arbitrarily chosen immediately succeeding or immediately past points). There is nothing special about the PV rate at the time at which the measurement happens to be made. Unlike the schedule variance of earned value, which at least is a cumulative measure and has been summed over all project activities, it is hard to see how the slope at that one point can legitimately characterize the project in any way; it doesn't even characterize the planning. At a minimum, one can argue that a longer time scale would at least average the noise in the data and give a better estimate of the PV curve's slope.
Although the formula will work for any functional shape of the PV curve, its assumption of a linear slope in the local region will generally not apply globally. Worse, the EV curve can assume any shape, as indicated by the jagged line in Figure 2. Because most project labor curves are conventionally S-shaped, the formula for ES, although it may work in special cases, has limited general utility.
Equation 1 also presents an arithmetical difficulty when using ES to assess performance. Measurement results should not be contingent upon the units in which the measurements are made. Unfortunately, the above m(i) is not the instantaneous slope at point i. It is instead an average slope, the difference between the two PV numbers at i and i + 1 (or i + 1 and i) divided by a single unit of time.
To understand the difficulty, consider the two points i and i + 14 on a PV curve, where i is measured in days. If the slope is constant over this interval, then any average slope calculated using any of the points between i and i + 14 will give the same answer. If ES is to be calculated at the end of day 14, then the slope calculation will use day 13. Now, suppose the i interval is measured in weeks instead of days, and the ES calculation is to be made at the end of week 2 (equivalent to day 14 with seven-day weeks). The slope calculation will now use week 1, which is equivalent to day 7 if the units are days. If the PV curve is not linear, then a slope calculated from P V(i + 14) – P V (i + 7) will not necessarily agree with one calculated over the much smaller one-day interval P V(i + 14) – P V(i + 13).
The standard schedule variance, calculated at only one point (e.g., day i + 14), does not change. However, the SV/m ratio does change, depending on the time units used, despite there being no differences between the two cases in either the performance or the plan!
Note that we are not worrying about the change in the units from days to weeks; that translation is handled correctly by multiplying or dividing the slope number by seven days per week and should introduce no inconsistencies. Thus, unless the planned value curve has a constant slope, different time units will give different answers. It is well-known that the planned value curve tends to take the famous S-shape, with the slope largest in the middle and approaching zero at the end. Therefore, as noted above, the slope of the PV curve changes over time.
In that case, if one changes the time interval for the measurement of ES, one can expect that the estimate of the earned schedule will change. From this point of view alone, ES has what one might term, at best, a certain inelegance. As a result, any numerical study of ES as a performance measure is suspect.
ES Undefined at the End of the Project
At the end of the project, the expression for Δ in Equation 1 becomes undefined; for example, we observe in Figure 1 that the planned value levels off before the earned value. Suppose we want to determine the value of the ES at t = 4 in Figure 1. At this point, the PV curve is flat, so the denominator of Δ is zero, and the expression is undefined.
ES itself is perfectly well defined at the end of the project—at any time, one can always draw the line from the earned value curve to the planned value curve, as shown in Figure 4. It is simply the expression for Δ that is undefined, yet another problem that derives from the linearity assumption in the expression for ES.
Figure 2: The standard derivation of earned schedule.
Fortunately, the method in the next section provides a solid foundation for the concept of earned schedule, and the estimation methods therein give much better estimates of the final schedule.
(Re)Defining Earned Schedule
Following Cioffi (2006), we begin by using upper-case letters, C (for “costs”) to denote the cumulative earned value quantities: Cb, where “b” represents “budgeted” (from “budgeted cost of work performed”) is earned value, and Cs, where “s” stands for “scheduled,” (from “budgeted cost of work scheduled,” more descriptive than “planned”) is planned value. Later, we will differentiate these with respect to time to describe their instantaneous forms, which will be denoted with a dot over the letter; again, both will depend on time. Because this whole discussion concerns schedule, we have no need to invoke the third element of the earned-value triplet, the actual cost.
We begin our exploration of schedule measurement by using a traditional S-shaped labor profile for the planned value. The Putnam-Norden-Rayleigh curve (Putnam, 1978), applicable to a variety of types of projects, produces an excellent S curve, and without loss of generality we can make the planned value equivalent to the completion rates of project activities. (If necessary, we can convert back and forth between completed activities measured in labor hours and the number of completed activities (Cioffi, 2006). This equivalence yields
where N, the total number of activities, is also the total planned cost, and tp represents the time of the peak in the planned labor rate curve, Ċs(t) (the derivative of Equation 4). We assume that the project is delayed, and so the earned value curve lags the scheduled cost expenditure:
where N, because of the assumption of no scope creep, is the same as the total planned cost in the previous equation. Figure 1 shows both S curves. In this equation, te represents the time of the peak in the earned value rate curve, Ċb, and te > tp.
With the same notation as in Equations 4 and 5 above, we write the instantaneous forms (Warburton, 2011):
Figure 3 shows both of these instantaneous curves.
After Lipke (2003), Stratton (2007), and Vandevoorde and Vanhoucke (2006) expanded the study of earned schedule. Their definition (Equation 1) is based on a graphical construction.
In this paper, we use Figure 3 to define earned schedule formally, without any approximations. We start with Lipke's graphical definition: to incorporate the measurement of a schedule delay (or acceleration) δ at time t, project back horizontally from that t point on the earned value curve to the
Figure 3: The instantaneous planned and earned value for the S shaped labor profiles in Figure 1. The profiles have the same area (i.e., the same total cost). Since te > tp, the project is delayed.
intersection of the planned value curve, which occurs at t – δ. Now, however, we will use S curves and not approximate the slope, and therefore our results should apply more generally.
Our formalism begins with the mathematical definition of the intersection:
From the above equation, one can see easily that for δ < 0, the schedule has been accelerated because the current earned value (i.e., at t), corresponds to a quantity that was to be obtained only later, at t = t + |δ|.
Using Equation 7 for the S curves in Figure 1, we have:
After canceling the N terms and taking the square root, we can find an expression for the delay:
where τ = tp/te (a ratio that, we will see, occurs often). From the definition derived graphically before, we now define the earned schedule as:
When the project is delayed, τ < 1, and when the project is accelerated, τ > 1. With the Putnam-Norden-Rayleigh formula for the S curve, this astonishingly simple but exact expression (to within the approximation of constant scope) for the delay, δ(t), shows not only that it is explicitly a function of time; but, interestingly, it grows linearly with time, as illustrated in Figure 4. In the next section, we examine a practical application of this formalism that demonstrates its great potential.
Figure 4: The schedule delay is the time difference between the two curves. With an S-shaped labor curve, the delay grows linearly in time.
Using the Schedule Performance Index to Find the Delay
The standard earned-value schedule performance index, here written as Is(t), is given by the ratio of the earned value to the planned value:
Early in the project, when t « tp, so that t/tp is small, the even smaller exponential argument allows us to write:
So, although under these (typical) S-curve assumptions, the actual curve for Is(t) varies with time (as shown in Figure 5), this early understanding of Is provides critical information by discovering τ: the evolution of the system is now determined! Also, because at any given time, t, in the project, t τ + δ = t; once we know τ, we also know δ (i.e., the delay), and from Equations 9 and 10, we know the evolution of both δ(t) and ΔS(t), the (new) earned schedule.
Estimating the Final Schedule
Once we know T, which can be found directly from the early time approximation above or from a later fit to the earned value's presumably S-shaped development (which yields te, with tp already being known from the start because of the planned value curve), we can also estimate the project's final duration.
The originally planned duration of the project is represented as ΔT1 (Cioffi, 2006). Let us assume the project has been delayed, and so now we have a new, unknown estimate of total project duration, ΔT′1 At the end of the project, the earned schedule equivalent, ΔS(ΔT′1) equals the original duration of the project, ΔT1, and so from Equation 10 we can write:
Figure 5: The Schedule Performance Index, I,(t), for an S-shaped labor curve. It begins at r2 and converges to 1.0 at the end of the project.
which we can solve to find the newly estimated duration, ΔT′1, which does not change as long as τ stays constant. Also note that, at any given time, 1/ τ yields the fractional delay in the project, which certainly holds true at project completion. Thus, we see that τ, easily determined from early project data, defines schedule performance, and the constancy of the estimate of the final schedule is also an inherent property of S-curves.
Although for the purpose of this initial discussion, we have assumed no scope creep (i.e., N is a constant), any scope changes would be easily handled. A new planned value curve would yield a new tp. If delays in the new plan occur, a new earned value curve would emerge, and thus a new τ, defining the evolution of the new schedule.
Summary and Project Management Applications
This work began when both of us, independently, saw problems in earned schedule as executed. Although we appreciated the validity of the concept, the troubling issue of a method dependent on the local slope of the planned value curve led us to develop the new definition we have detailed above.
We have created a more precise, formal definition, which, with the quite-reasonable assumption of the classic S-curve growth, results in two extraordinarily simple but powerful formulas. The first formula, Equation 10, shows that the earned schedule, ΔS, is directly proportional to time, and it scales according to the parameter τ that describes the S curves for planned and earned value. The second, Equation 12, shows that at early times, the standard schedule performance index is directly proportional to τ2. (If necessary or desired, one can also make a later determination of τ from an S-curve fit to the earned-value data to find its characteristic time, te; tp comes from the planned value.)
Under the assumption of constant scope, τ remains constant: once you have obtained it, you know the schedule delay as a function of time for the entire project. If the scope should change, a new planned-value curve will result in a different tp and so, τ is easily adjusted.
In future work, we will examine other types of work profiles (e.g., a linear one as well as at least one other formulation of the S-curve) (Cioffi, 2005). Our initial investigations of these suggest that the same linearity applies.
Such an extension of the general applicability of the technique (i.e., that it does not depend on the details of the mathematics of the S curve), would deepen the result that one can find a firm prediction of ultimate schedule delays based on early earned value numbers. Unlike the standard schedule performance index, which can change, the system's evolution is determined once τ has been found. In short, in the spirit of the standard earned value indices of cost and schedule, we have found another useful index.