PERT Completion Times Revisited

1. Introduction

Project management is a staple in introductory operations management or management science courses. Not only are project management concepts and methods important in practice, but the topics readily lend themselves to an array of useful models built on simple concepts that can be easily understood and mastered at the introductory level. For at least these reasons, network planning models such as PERT and CPM are very natural entrees to the world of model building and analysis.

Introductory discussions usually begin by explicating the basic concepts of activities, durations, and precedence relationships, followed by the development of network representations of a project; earliest and latest start and completion times; slack; and critical path(s). Attention soon moves to modeling projects with significant randomness – where activity durations are not deterministic, but are random. The classic PERT model then unfolds, with independent activities; optimistic, pessimistic, and most likely time estimates related to beta distributions; and the PERT approximations.

Attention focuses on finding the expected duration and variance of the critical path, and with an appeal to the central limit theorem, using the properties of the critical path duration to make probability statements about project completion. Many authors warn readers of the limitations of these probability statements – the danger of near critical paths becoming critical, departures from the normal distribution assumed, etc. Some also mention that simulation would be a more apt analytical method of estimating accurate probabilities. However, despite the usual disclaimers and admonitions, the uninitiated reader is likely to leave a typical introductory PERT discussion with a clear impression that PERT yields useful, if only approximate, results. This paper illustrates some risks inherent in such an impression by contrasting PERT estimates and spreadsheet simulation results for some relatively simple examples. We hope these examples motivate increased pedagogical emphasis on (a) the tenuousness of PERT estimates and (b) the merits of spreadsheet simulation in project management.

2. PERT Completion Times: The Standard Approach

To facilitate exposition, we begin with a brief summary of the PERT approach, using the example in Figure 1 (from chapter 3 of Heizer and Render (2005)), concerning the installation of a pollution control system.

Figure 1 shows the project data, and calculation of the critical path and its mean and variance, using the methods of Ragsdale (2003) and the usual PERT approach. [Click here to download the Excel file for this example.] The critical path is ACEGH with expected duration E(D) = 15 and variance(D) = 3.11. Appealing to the central limit theorem, the PERT method approximates the probability distribution of D with the normal distribution, N(E(D),stdev(D)) = N(15,1.76), and uses this approximation to make probability statements about project duration.

Example 1: Milwaukee General Hospital

Figure 1

Figure 2 shows the precedence diagram and typical computational results.

Milwaukee General Hospital Precedence Diagram

Figure 2

At this point, most discussions note a major limitation of the approximation – its myopic focus on the critical path ignores other near critical paths that might turn out to be the actual critical path(s) upon execution of the project. Two likely candidates in our example are ADGH or BDGH. Ceteris paribus, with a slack of one (1), an unfortunate time for activity D could push ADGH or BDGH to criticality. Section 3 provides examples of this and other situations in which near critical paths become critical.

Most authors mention that the complexities posed by multiple critical paths (along with some other departures from the basic PERT assumptions) are most appropriately and readily addressed by simulation models, although relatively few introductory expositions – especially in operations management texts – ever return to address the specifics of simulating projects with important random characteristics. Some introductory OR/MS texts do explicitly address the simulation approach (see, for example, Hillier and Hillier (2003) or Ragsdale (1998)).

Other possible limitations of the PERT approximation include:

· Activity times are not stochastically independent.

· The critical path comprises fewer activities than reasonable application of the central limit theorem requires. Our example has a five-activity critical path, far short of the typical n ≈ 30 rule of thumb for the central limit theorem.

· The PERT approximations of activity duration mean and variance can deviate significantly from reasonable and accurate approximations of the underlying activity duration probability distribution.

To our knowledge, the first is relatively unexplored, perhaps because dependent activities would significantly compound model complexity. The second is rarely mentioned, probably because (a) it is so obvious and (b) many, if not most, actual projects will entail enough activities to justify using the central limit theorem. As for the third, while MacCrimmon and Ryavec cited misspecified activity probability models in their early analytical study of the PERT assumptions (1964), introductory discussions rarely mention this error source. This is unfortunate since, as the examples in sections 3 and 4 will show, the PERT beta approximations can also compound inaccuracies.

3. Near Critical Paths Can Turn Critical – And Often Do: Some Examples

We begin with two variants of the Milwaukee General Hospital (MGH) project:

MGH ~ Initial MGH parameters (Figure 1)
MGHB ~ A slightly modified MGH formulation, with parameters

Activity B: Change (a,m,b) from (2,3,4) à (1,3,10)
Activity D: Change (a,m,b) from (2,4,6) à (1,3,10)
Activity F: Change (a,m,b) from (1,2,9) à (3,6,18)
All other parameters remain unchanged.

The changes in 2.a-c do not affect the critical path, so the same PERT approximations of the expected project duration and variance apply to both variants. However, the changes simultaneously reduce the slacks of B, D, and F and increase the variance of each, thus increasing the likelihood that activities B, D, or F will become critical.

To assess the adequacy of PERT approximations in this example, we compare three CDFs of project duration:

PERT approximation
Empirical CDF from a 100,000-trial Crystal Ball (2005) simulation of MGH
Empirical CDF from a 100,000-trial Crystal Ball simulation of MGHB.

Figure 3 contains these three project duration CDFs:

Project Duration CDFs for Two Milwaukee General Hospital Variants

Figure 3

We regard the simulated results as baselines for their respective variants. Note that the PERT approximation does not estimate either baseline very accurately. For the initial MGH formulation, PERT overestimates F(d) for shorter durations (below 13.6) and underestimates F(d) for longer durations (above 13.6). PERT also consistently and significantly overestimates F(d) for the modified version, MGHB. While an eyeball test is probably a reasonable standard of the quality of fit here, Table 1 contains two common quantitative measures of fit, the mean absolute deviation, MAD, and the mean absolute percent error, MAPE.

PERT Approximation Errors: MGH Simulations

Table 1

In other words, the average absolute deviation between the PERT approximation and the MGH simulation CDF is .059. In a similar fashion, PERT overestimates F(d) by an average of .117 for MGHB (since all deviations are overestimates). The MAPE values in Table 1 are perhaps slightly overstated, since small errors in the left tails of the distributions generate inflated absolute percent errors. Nonetheless, suffice it to say the PERT approximations do not accurately estimate the project duration CDFs.

As expected for the reasons discussed in Section 2, multiple critical paths emerged in the simulation trials. Two main paths surfaced in the initial MGH and four emerged in MGHB, with the approximate frequencies in Table 2.

Critical Paths in MGH Simulations

Table 2

One might wonder if these results are unusual. A second example suggests (but of course does not prove) otherwise. Figures 4-6 contain the second example (from chapter 8 of Krajewski and Ritzman (2005)), concerning the relocation of a hospital.

Example 2: St. Adolf’s Hospital (SAH)

Precedence Diagram

Figure 4

Example 2: St. Adolf’s Hospital

Figure 5

Figure 4 is a precedence diagram and standard computational results for St. Adolf’s Hospital (SAH). Bold borders identify the critical path, BDHJK. Double lined borders identify a second path, ACGJK, which is near critical.

Figure 5 shows a Crystal Ball spreadsheet simulation model for SAH. [Click here to download the Excel file for this example.] Standard PERT estimates yield an expected length of 69 weeks (cell R20) and a variance of 11.889 (S21). Note that the actual variance of the duration of BDHJK, calculated from the underlying beta distributions of the activity durations, is also 11.889 (X21). The actual expected duration of BDHJK, calculated from the underlying beta distributions of the activity durations, is 69.3 (I20 or W20), slightly above the PERT estimate of 69.

Figure 6 shows three SAH project duration CDFs – the standard PERT approximation, N(69,3.45); the empirical CDF from a 50,000 trial simulation; and a PERT adjusted approximation, N(66.3,3.45). As was true for the first example, MGH, the PERT approximation is not a very close fit to the baseline simulated project duration CDF. Moreover, the PERT adjusted approximation is not much better.

Project Duration CDFs for St. Adolf’s Hospital

Figure 6

Table 3 contains the MAD and MAPE values for this example.

PERT Approximation Errors: SAH Simulations

Table 3

The results in Figure 4 suggest that in addition to the critical path, BDHJK, the near critical path, ACGJK, is likely to turn critical in any realization of the project. Indeed, as Table 4 shows, ACGJK turned critical in 1/3 of the trials.

Critical Paths in St. Adolf’s Hospital Simulations

Table 4

As the three foregoing examples clearly demonstrate, PERT approximations can offer less than adequate estimates of project duration. It is worth noting that we borrowed these two simple examples, Milwaukee General Hospital and St. Adolf’s Hospital, from excellent, representative, introductory discussions. By no means were they carefully chosen to illustrate the approximation errors displayed in this section.

It is also probably worth mentioning that most introductory discussions suggest that PERT estimates are optimistically biased, tending to uniformly overestimate the CDF of project duration. This plausible property seems like a natural consequence of near critical paths turning critical. This impression is not limited to introductory discussions. In the abstract of his early paper on this topic, A. R. Klingel (1966) says, “Among network techniques recently widely employed in program management, Pert is addressed to the problem of assessing the manager’s chances of completing a project on time. Theory and monte carlo simulation have shown that the Pert method yields results that are biased high,…” While Klingel’s assertion might be true of larger projects, the results for our simple examples suggest the bias can cut both ways – positive or negative.

4. Activity Times, Beta Distributions, and PERT Approximations

It is useful to clarify the activity time probability models implicit in PERT approximations. We start with a brief overview and then turn to a slightly more extended (and technical) discussion. On one hand, the second discussion seems like overkill, yet it also seems necessary in order to explicate clearly the various issues involved.

Introductory PERT discussions deal with random activity durations by first introducing the concepts of three time estimates: optimistic, most likely, and pessimistic. There inevitably follows an intuitive discussion relating these three estimates to the form of the assumed underlying probability distribution of the activity duration, posited to be a beta distribution. After briefly exploring the flexibility and suitability of the beta distribution, discussion quickly moves to the standard approximations of activity mean and variance:

· D = duration of the activity

· E(D) = expected duration = (a+4m+b)/6

· Variance(D) = ((b-a)/6)²

where a, b, and m are, respectively, the optimistic, pessimistic, and most likely times for the activity. The activity means and variances are used to compute various intermediate variables (EST, EFT, LST, LFT, slack), which in turn help identify the critical path. The expected duration and variance of the critical path are then calculated.

In sum, PERT offers the beta as a plausible probability model of activity times, relates the three usual estimates, (a,m,b), to properties of the beta distribution, and translates (a,m,b) into approximations for the mean and variance of each activity. Interestingly, PERT pays little or no explicit attention to precisely which beta distribution is being proffered, presumably because only the mean and variance are used. While that shorthand serves PERT’s purposes well, it falls short of providing the level of detail required, say, to conduct a simulation of the project. That would require a specific beta distribution for each activity.

In fairness, in their seminal PERT paper, Malcolm, Clark, Roseboom, and Fazar (1959) addressed a comprehensive system in which the individual activity times and their probability models played an important, but secondary role. Arguably, they adopted approximations entirely appropriate and adequate to the context in which they were developed. In fact, Clark (1962) explicitly addressed these issues, clearly explaining the complex setting in which “estimates must be made periodically, formally, and at low cost for thousands of activities.” He went on to explain the natural choice of β-distributions absent a compelling alternative; the rationale for selecting the extremes and mode as the basic parameters to be subjectively estimated; and the choice of fixing a specific beta by assuming 1/6 of the range to be the standard deviation. He then noted that thus fixed, the final step of converting the subjective inputs to a specific β-distribution “requires computations, including the solution of a cubic equation, which are ponderous relative to the reliability of the results,” and concluded with a rationale for the standard PERT approximation of the mean.

Although Clark’s rationale and explanation are clear (albeit not expressed in a mathematically explicit form), the issues surrounding beta distributions and PERT approximations prompted others to pursue the topics in more depth, beginning soon after the seminal PERT paper and extending through the late 1980s. These efforts produced numerous refinements to translate subjective estimates of parameters into specific beta distributions. Keefer and Verdini (1993) summarize and numerically assess the accuracy of these methods for a range of β distributions. Five of these methods offer precise linkages between the usual subjective estimates (a,m,b) and the parameters of a specific beta distribution for an activity time. We summarize those linkages very succinctly below, after a brief summary of the beta distribution and some of its summary statistics. We do not consider some other methods because, while interesting and potentially useful, they require inputs other than (a,m,b) – usually other fractiles of the distribution – not available in our examples.

The beta distribution is a two parameter continuous distribution on the open real interval (A,B) (Some sources use the closed interval, [A,B].) The density function is

See, for example, NIST/SEMATECH e-Handbook of Statistical Methods (2005).

α and β are shape parameters, and A and B are the minimum and maximum values, respectively. The case A = 0 and B =1 is called the standard beta distribution. If X follows a general beta distribution with parameters α and β on (A,B), A is also a location parameter, and A-B is a scale parameter. This is most easily seen by noting that X, on (A,B) is related to the standard beta distribution Y (with the same parameters α and β) by the transformation X = (B-A)Y + A.

Table 5 lists some common statistics for the standard and general beta distributions.

Statistic	Standard beta	General beta
Range	(0,1)	(A,B)
Mean
Mode α,β>1
Variance

Some Summary Statistics for the Beta Distribution

Table 5

With this terminology, we can briefly sketch details of five estimation procedures for explicitly translating the (a,m,b) estimates into a specific beta distribution.

The procedure implicit in Malcolm et al (1959) and made explicit by Clark (1962) amounts to the following:

M1. Obtain the estimates (a,m,b).

M2. Fix the mode by imposing equation 1.

M3. Fix the variance by imposing equation 2.

M4. Solve equations 1 and 2 for α and β, which fixes the specific beta distribution.

1. [b(α-1)+a(β-1)]/(α+β-2) = m Mode

2. αβ(b-a)²/[(α+β)²(α+β+1)] = [(b-a)/6]² Variance

Equations 1 and 2 yield the cubic equation that Clark mentioned in the foregoing quote. In the symmetric case, m = (a+b)/2, the solution is α=β=4. In the asymmetric case, the cubic is indeed ponderous and does not seem to yield a closed form solution even for the standard beta distribution (Interested readers can contact the author for detailed Maple results confirming the intractability of these equations. On the other hand, equations 1 and 2 yield readily to numerical solution with a tool like Excel’s Solver.)

Grubbs (1962), roundly criticizing the Malcolm et al PERT assumptions, argued that the traditional estimates for the mean and variance, (a+4m+b)/6 and ((b-a)/6)², respectively, overly constrain the parameters of the beta distribution and restrict it to “one of three fat, flat Beta distributions” (1962b). While internally consistent, Grubbs’ method is curious in that he approached the estimation process in a manner subtly, but distinctly different from that adopted by Malcolm et al. Grubbs offered the following:

G1. Obtain the estimates (a,m,b).

G2. Fix the mean by imposing equation 3.

G3. Fix the variance by imposing equation 2.

G4. Solve equations 2 and 3 for and β, which fixes the specific beta distribution.

3. (αb+βa)/(α+β) = (a+4m+b)/6 Mean

Equations 2 and 3 solve readily, yielding the following unique solutions:

4. α = β = 4 Grubbs symmetric

5. α = 3 – SQRT(2) β = 3 + SQRT(2) Grubbs positively skewed

6. α = 3 + SQRT(2) β = 3 – SQRT(2) Grubbs negatively skewed

These results lead Grubbs to conclude that the PERT assumptions “limit us to one of three fat, flat, Beta distributions”. Moreover, these parameter values yield one of three modes:

7. (a+b)/2 Grubbs symmetric mode

8. (a+b)/3 – (b-a)SQRT(2)/6 Grubbs positively skewed mode

9. (a+b)/3 + (b-a)SQRT(2)/6 Grubbs negatively skewed mode

In the symmetric case, the Grubbs mode is precisely the original subjective estimate m. However, the asymmetric cases 8 and 9 exhibit a curious property – the calculated modes are not equal to the original subjective estimates m. This underscores the subtle, but important distinction between the approaches of Grubbs and Malcolm et al in their seminal paper. Both valid approaches address related, but subtly different, problems.

Donaldson and Coon (1964) take still another tack in closely related papers. Donaldson bases his estimates on subjective estimates of the optimistic, pessimistic, and mean times, and the assumption that the β-density f_X is tangential to the horizontal axis at the extremes, a and b. Coon extended Donaldson’s method to handle PERT estimates, (a,m,b). A brief sketch of Coon’s method follows:

CD1. Obtain the estimates (a,m,b).

CD2. Fix the mode by imposing equation 1.

CD3. Impose the tangency assumption, which is equivalent to α>2, β>2.

CD4. Equation 1 and condition CD3 and do not uniquely fix α and β, but define a family of β-distributions (see Coon’s comments below).

CD5. Fix the distribution by finding the smallest sum, α+β, satisfying equation 1 and CD3. Since Variance = αβ(b-a)²/[(α+β)²(α+β+1)], the resulting distribution has the largest variance among those that satisfy equation 1 and condition CD3. The resulting α and β values are given in equations 10-12.

10. If b-m > m-a α = 2 β = 2(b-m)/(m-a)+1 CD positively skewed

11. If m-a > b-m β = 2 α = 2(m-a)/(b-m)+1 CD negatively skewed

12. If b-m = m-a α = β > 2 CD symmetric

Neither Donaldson nor Coon explicitly addressed the symmetric case, but applying their logic would lead to the results in 12 as estimates for the symmetric case. (We have used α=β=4 in the simulations MGH, MGHB, and SAH.)

The Coon-Donaldson estimates have an interesting property:

13. If α and β satisfy Coon’s conditions, so do α' and β’, where

α' = α + δm/(1-m) δ>0

β’ = β + δ δ>0

A similar property (with a slightly different form) exists for Donaldson’s estimates. In other words, the Coon-Donaldson estimates generate families of beta distributions, rather than unique distributions. In fact, Coon concludes her paper with the following comments (here x₁=a, x₂=subjective estimate of the mean, and x₃=b):

It should be clearly pointed out that estimates of the end points and the mean (or mode), even when coupled with the assumption that the β-distribution is tangent to the x-axis at both ends, does not lead to complete generalization of PERT activity time distributions. In effect, what Donaldson’s method does is to set β=2 for all curves where the mean is less than one-half the range of the curve and then to estimate α from the ratio

β/α = (x₃- x₂)/(x₂- x₁)

On the other hand, if the mean is greater than one-half of the range, then α=2 and β is determined from the β/α ratio. Hence we are still left with a restricted set of β-distributions, although the current restrictions have been greatly relaxed by allowing the distributions to take on varying degrees of skewness as compared to the severe restrictions pointed out by Grubbs.

Farnum and Stanton (1987) explore conditions that justify the PERT estimates, and as a byproduct, develop improved estimators for those cases in which the standard estimates are poor. In the course of their analysis, they suggested the following estimation procedure to translate the usual estimates (a,m,b) into a unique beta distribution:

FS1. Obtain the estimates (a,m,b).

FS2. Fix the mode by imposing equation 1.

FS3. Fix the variance by imposing equation 14.

FS4. Fix the distribution by solving equations 1 and 14 for α and β.

14. (α-1)(β-1)(b-a)²/[(α+β-2)²(α+β-1)] = [(b-a)/6]² Variance(α-1,β-1)

Note that the RHS of equation 14 is not Variance(α, β), but Variance(α-1,β-1), an approximation Farnum and Stanton briefly justify, without providing much in the way of detailed substantiation. At any rate, the net result of equations 1 and 14 is the following closed form expressions for α and β in 15 and 16:

15. α = [36[((m-a)/(b-a))² + 1](b-m)/(b-a) Farnum-Stanton

16. β = [36[((b-m)/(b-a))² + 1](m-a)/(b-a) Farnum-Stanton

Golenko-Ginzburg (1988) proposed the following refinement, based on yet another set of relaxed assumptions:

GG1. Obtain the estimates (a,m,b).

GG2. Fix the mode by imposing equation 1.

GG3. Assume α+β = z = constant. Golenko-Ginzburg justifies this condition as an extension of earlier assumptions, saying, On the basis of statistical analysis and some other intuitive arguments, the creators of PERT assumed that p+q ≈ 4. (Golenko-Ginzburg defined p = α -1 and q = β -1.)

GG4. Using the standardized completion time, Golenko-Ginzburg calculated the variance of the completion time, using the subjective estimate, m, to be σ²(m)=(1+z+z²m-z²m²)/((z+2)²(z+3))

GG5. Fix z by requiring that the average of σ²(m) over all values, 0<m<1, is 1/36, or equivalently, ∫σ²(m)dm = 1/36.

GG6. Golenko-Ginzburg did not find α and β, but directly calculated the mean and variance of the general completion time, X, in expressions 17 and 18.

GG7. Fix the mean by imposing equation 19.

GG8. Fix the distribution by solving equations 1 and 19 for α and β.

17. mean(X) = (2a+9m+2b)/13

18. variance(X) = (b-a)²[ 22 + 81(m-a)/(b-a) – 81{(m-a)/(b-a)}²]/1268

19. (αb+βa)/(α+β) = (2a+9m+2b)/13 Mean (Golenko-Ginzburg)

Equations 20-22 show the Golenko-Ginzburg expressions for the asymmetric case.

20. α = ρβ where ρ = [9(m-a)+2(b-a)]/[9(b-m)+2(b-a)] GG asymmetric

21. α = ρ[(b-m)-(m-a)]/[ρ(b-m)-(m-a)] GG asymmetric

22. β = [(b-m)-(m-a)]/[ρ(b-m)-(m-a)] GG asymmetric

For the symmetric case, m = (a+b)/2, so equation 1 yields α=β. Solving equations 1 and 19 simultaneously yields 23:

23. α + β = 6.5

Substituting α=β into equation 23 and solving yields, for the symmetric case:

24. α = β = 3.25 GG symmetric

Table 6 contains the resulting numerical values for nine representative distributions arising in the previous examples. Included are the estimates (a,m,b), the PERT variance ((b-a)/6)², and the values of the values of α and β resulting from each method. The calculated mode is included for Grubbs’ method since it does not faithfully reproduce the subjectively estimated mode. We include calculated variances for Coon-Donaldson, Farnum-Stanton, and Golenko-Ginzburg, since Coon-Donaldson does not assume a variance and neither Farnum-Stanton nor Golenko-Ginzburg faithfully reproduces the assumed PERT variance, ((b-a)/6)².