New Tools for Project Managers: Evolution of S-Curve and Earned Value Formalism

Transcription

1 New Tools for Project Managers: Evolution of S-Curve and Earned Value Formalism A Presentation at the Third Caribbean & Latin American Conference On Project Management, May 2003 Denis F. Cioffi, Ph.D. Director, Project Management Program, The George Washington University, Monroe Hall 302, 2115 G Street NW, Washington, DC Abstract The so-called S-curve of a project s accumulated costs is given a closed mathematical form that allows specification of a slope and the time at which half the total funds are spent. In addition, a more compact, uniform, and ultimately clearer notation for earned value analysis is presented. 1 Introduction: Giving the S-Curve Concept a Mathematical Basis I like teaching and studying project management because of the strong mix of science and art. More specifically, proper project management practice demands expertise with skills both technical and emotional. In terms of its importance to the success of any one project, the human side of management almost always trumps the numerical side. However, the progress of project management as an independent discipline requires continued improvement in the scientific approach to managing projects. In this article I show improved notation for earned value analysis and a formal equation for the S-curve seen in typical projects. address: [email protected]. URL:

2 To understand this curve, we begin with an example of its derivative, as seen on a graph of project cost per unit time versus time. Figure 1 shows data taken from an example project in Version 8 of Scitor s Project Scheduler software [6], where the unit of time is the day. The data points show the daily costs, in dollars, for a 207 calendar day project PS8 intranet Data 2000 Cost (Dollars) Per Day Time (Days) Fig. 1. The solid curve is an (arithmetically calculated) derivative of an S-curve fit, and so its units are also in dollars per day. Because many days (e.g., weekends) have zero cost, the solid curve on the graph shows a smaller magnitude than many of the daily costs, but it legitimately represents the average cost per day throughout the project. When we take the integral of this curve to sum the costs to each time, we produce the traditional project management S-curve of accumulated cost, as shown in Figure 2. We are comfortable speaking of this s-shaped curve, but does it have a definition? The Guide to the Project Management Body of Knowledge (the PM- BOK) defines the curve in terms of its appearance: only: Graphic display of cumulative costs, labor hours, percentage of work, or other quantities, plotted 2

3 against time. The name derives from the S-like shape of the curve (flatter at the beginning and end, steeper in the middle)... [5] PS8 Intranet Example y Fig. 2. The solution, Equation 1 (dashed line), fit to sample data (jagged solid line). The two parameters fit are γ = and r 0.67 =0.746, with a resulting β 1/2 = The (Pearson s) coefficient of determination, R 2, equals Note that the equation forces y =1atβ =1. As the example shows, and as the PMBOK mentions, this plot arises when cumulative project costs are graphed against time. The PMBOK also mentions that one can plot cumulative labor hours, and indeed, such graphs are often used with descriptions of project phases. They are often shown incorrectly, however, which calls for a brief aside on the definition of effort. Effort, E R, is the product of the duration of an activity, t, and the particular resource being used at some specific intensity, R I : E R R I t. For example, one may speak of 2 workers (resource: worker; intensity: 2) need- 3

4 ing two hours for a task (resource duration: 2 hours) for an effort of 2 workerhours. (As explained in Managing Project Integration [2], the more sophisticated version of this equation makes explicit that the duration itself is a function of the resources used.) Only efforts with identical dimensions can be added to each other. For example, adding lawyer-hour effort to worker-hour effort makes no sense. However, effort is converted to cost by multiplying it by the resource cost rate (cost per unit time per unit resource). In the example above, a cost rate of so many dollars per worker-hour would yield the cost in dollars. Thus, instead of an improper labor-hours curve with different resources combined, costs obtained from the efforts of many different resources can be added to produce a proper accumulated cost S-curve. 2 An Analytic Parameterization of the S-Curve What project managers call an S-curve, many others know as the logistics curve or the curve of population growth. It results from the solution to a differential equation (that I encountered, for example, in a discussion of mathematical stability [8]) and can be used also to show the growth of interstellar hydrogen clouds in the spaces between the stars (the interstellar medium) [4]. As I describe more completely in a preprint that sits now on my website [3], I first changed the generic solution s boundary conditions to make them more appropriate for project management. To make the equation easier to use, I then divided the curve into three parts and approximated the slope in the middle section; the project manager chooses the parameter r 0.67 to fix the slope (it can equal 1). I also normalized the equation to total project cost and duration, the latter through β = t/t 1, where t 1 is the project duration. The project manager chooses the approximate time at which half the total cost has been spent, β 1/2. (The website paper shows how to calculate β 1/2 so the half-way point is exactly where desired.) The final result is y(β) y 1 e 8 r 0.67 β 1+γ e 8 r 0.67 β, (1) where y is the cost normalized to the total cost, γ is a constant defined through ln (γ +2)=8r 0.67 β 1/2, (2) 4

5 and y is given by y 1+γ e 8 r e 8 r (3) By definition, y = 1 at the end of a project and y y > 1 beyond that time. At β = β 1/2, y = y /2. Figure 2 shows the excellent fit 1 of this equation to the same data shown in Figure 1. Approximation at Three Values of r r 0.67 = y β 1/2 = β Fig. 3. The solution (Equation 1) shown with β 1/2 =0.5 for three values of r How To Use This Solution After the development phase, but before execution, the proposed spending profile of a well-planned project results from the accumulation of the specific costs 1 Via the general curve-fit function of Kaleidagraph software, Version

6 estimated for all scheduled deliverables. After execution, the actual S-curve is produced from expenses. However, before development, when one wishes to forecast accumulated expenditures of the potential project, this equation would be useful. As shown in Figure 3, one can adjust the rise (r 0.67 ) to produce different types of cost curves. Similarly, one can also adjust the halfway point (β 1/2 ) (see the full work, [3]). Future work may show typical values of r 0.67 and β 1/2 for projects under different conditions, and then project managers and senior executives could make more accurate forecasts. 4 Improved Earned Value Notation The S-curves above lead rather naturally into the topic of earned value because the plots typically used to demonstrate earned value concepts are (partial) plots of accumulated costs. In Managing Project Integration [2], I presented an improved earned value notation that I condense and supplement below. The above treatment of S-curves aims to take a verbal concept and put it on a numerical basis. Now, this recasting of the familiar earned-value parameters into a foreign appearance attempts to raise their notation to a place where analysts can study them more abstractly and practitioners can use them more easily. When I began teaching project management, I found that I did not like manipulating four-letter abbreviations in my head. I started playing with a new notation, with little hope of it ever moving beyond my students. However, when the Project Management Institute revised the abbreviations (making them shorter but for the most part, less clear), I decided to offer my notation, too. Below are the transformations of the budgeted cost of work performed, the budgeted cost of work scheduled, and the actual cost of work performed: Earned value (BCWP): C b Planned value (BCWS): C s Actual cost (ACWP): C a The revised official terminology precedes the new symbols, which may be read easily as budgeted cost, C b ; scheduled cost, C s ; and actual cost, C a. The first advantage of this notation arises with the definition of the cost differences (a simpler, more precise word than variance ) between expected costs 6

7 Behind Schedule and Over Budget Cost (Normalized) Fraction of Planned Total (C ) s, Earned value parameters should be measured early! C a C s At any given time, measure earned value cost and schedule differences vertically. C a = C b - C a 0 C b C s = C b - C s β= t/t 1 Fig. 4. The three primary earned value parameters at 25 percent of a project s planned total duration. Because here the earned value (the budgeted cost of work performed) is less than both the scheduled and actual costs, the resulting negative differences, C a and C s, show this project to be behind schedule and over budget. and actual costs in both schedule and budget. Both equations can be expressed compactly via: C i C b C i, (4) where i is set to a for cost and s for schedule. If the difference is negative, the project is behind schedule (when i = s) or over budget (when i = a). Following a comment made by my colleague at The George Washington University (and co-presenter at this conference), Dr. Frank Anbari [1], I use the inverse of the typical index ratios. Then, rather than call them indexes, I term them cost and schedule performance factors. Because these factors will multiply planned costs and durations, numbers less than one show performance better than planned, and numbers greater than one show underperformance. 7

8 They can again be defined with one equation: F i C i C b, (5) where i is set to a for the cost factor and s for the schedule factor. Cumulative cost factors can be defined as a sum of individual factors multiplied by their corresponding weights (see [2]), and where possible, they should be used in the calculations of future values. With the notation presented here and in Managing Project Integration [2], estimates of future project durations and costs based on current durations and costs are made simply through multiplication by the appropriate factors (hence the name). If we use the subscript 1 to denote the end of a project, its total duration can be written as T 1, and the earned value estimate of the new duration, denoted with a prime, is given by: T 1 = T 1 F s. (6) The so-called budget at completion is written as C s,1, and any new estimate of this quantity will be C a,1, where the prime reminds us that the number is a prediction, but the a reinforces the hope that we are calculating an actual cost. It is not a scheduled cost because it has not been derived directly from a cost curve. If all costs were recalculated and rescheduled because of the information gained in an earned value analysis, one would have a re-baselined project with a new C s,1. Fleming and Koppelman [7] cite three ways of using earned value to estimate a project s final cost. They observe that some have called the overrun to date estimate useless. It does not take advantage of past performance in its look to the future, but merely adds the existing difference between the planned total cost and the earned value (C s,1 C b ) to the current actual cost (C a ). Silly. The new notation reveals that another possible method, the high-end cumulative CPI times SPI EAC, is a strange combination of schedule factors, actual costs, and the proper estimate: C a +(C s,1 C b ) F s F a = C a,1 F s (F s 1) C a. The proper estimate (the low-end cumulative CPI EAC ) is shown straightforwardly here, as expected: C a,1 = C s,1 F a. (7) In Managing Project Integration [2], I also suggest a possible combination 8

9 factor, the cost-schedule performance factor: Φ F a + F s (F a F s ) 2. (8) If greater than one, this number indicates problems in at least one area, schedule or cost. 5 Summary Those who study project management and profess it to be an academic discipline have a duty to add depth to its portrayal as both a science and an art. Where the tools of science will increase our ability to manipulate quantities necessary for good management and therefore increase our understanding we should embrace those tools. I have presented an equation that transforms the verbal description of the typical project management S-Curve into a precise mathematical formula, and I have shown a notation that can increase the clarity of earned value analysis. Together, these help turn earned value analysis into earned value management. References [1] Frank Anbari. An Operating Management Control System for Large-Scale Projects. In American Institute for Decision Sciences, Ninth Annual Meeting, Northeast Regional Conference, Philadelphia, [2] Denis F. Cioffi. Managing Project Integration. Management Concepts, Inc., Virginia, [3] Denis F. Cioffi. A tool for managing projects: An analytic parameterization of the S-Curve. To be submitted to the International Journal of Project Management, [4] Denis F. Cioffi and J. Michael Shull. Simulations of supernovae-dominated interstellar media in disk galaxies. The Astrophysical Journal, 367:96, [5] Project Management Institute Standards Committee. A Guide to the Project Management Body of Knowledge 2000 ed. (PMBOK). Project Management Institute, Inc., [6] The Sciforma Corporation. Project Scheduler software, PS8. ( suite/ps8 overview.htm),

10 [7] Quentin W. Fleming and Joel M. Koppelman. Earned Value Project Management. Project Mangement Institute, Pennsylvania, second edition, [8] Ilya Prigogine. From Being to Becoming: Time and Complexity in the Physical Sciences. W.H. Freeman and Company, San Francisco,