# Project Management Estimation Methods – PERT (Program Evaluation and Review Technique)

Posted by mgocean on May 26, 2009 in Estimation, Project Management |

PERT is probabilistic and stochastic in that variability is considered in the duration of each activity. It means PERT assumes that the duration of each activity is represented by a random variable with a known probability density function. (Lee D.E., 2005, pp: 310 – 318)

The PERT technique uses additional information from the estimator to produce a better estimate. The estimator provides three estimates for the quantity of interest: the lowest possible value, the most likely value, and the highest possible value, denoted by L, M, and H, respectively. (The basic mathematics assumes that these estimates are unbiased, and that L and H correspond to the minus and plus three-sigma limits of the distribution.) Using these three values, the PERT technique calculates the expected value, E, and the standard deviation, σ, using the following equations:

E = (L + 4 * M + H)/6

σ = (H – L)/6

“E” represents the estimate of the average value. These equations are very simple to apply in practice, leading to widespread use by estimators. Even if the underlying assumptions are not exactly satisfied, the technique does provide some indication of the degree of uncertainty in the estimator’s mind. The key indicator is to look at the ratio σ/E. (Statisticians call this ratio the “coefficient of variation.”) A large value of this ratio indicates a high degree of uncertainty on the part of the estimator.

You can easily implement the PERT technique using a spreadsheet. All items on a worksheet must have the same units of measure if you intend to calculate a total and its standard deviation. (Prepare a separate worksheet for each set of items having the same units of measure.) For example, to estimate the sizes of a set of software modules, you would list each module on a separate row of the spreadsheet. The columns of the spreadsheet would be the module’s name, the lowest value, the most likely value, the highest value, the expected value (the mean), the standard deviation (σ), and the ratio of (σ/E). Figure 4‑10 shows an example with four modules, identified in the left column. The next three columns contain the three estimated size values for each module. The last three columns show the values calculated by the PERT equations. Use the following formulas to calculate the total value and the standard deviation of this total:  Figure 4‑10 Example of PERT technique results

These equations assume that each estimate is independent of the others. That is, they belong to different random distributions. This means that the variance of the total value is computed by adding the variances of each estimate, computed using the preceding formula. The standard deviation is then the square root of the total variances. Do not compute the standard deviation of the total by summing the columns to obtain Ltotal and Htotal, and then using these values in the equation for σ.

If you can estimate the mean, μ, and standard deviation, σ, you can estimate the degree of risk involved in an estimate. (One way to obtain μ and σ is to use the PERT technique.) Assume that the error in the total size follows a normal distribution as shown in Figure 5-5. The peak of this distribution lies at the expected value, E. The width of the curve is determined by the standard deviation, σ. Set the bid value, B, to the expected (mean) value, μ, plus some multiple, t, of the standard deviation, σ: B = μ + t*σ. If t = 1, B = μ + σ. For a one-tailed normal distribution, the probability that the actual value will lie below the value B is 0.84 (= 0.50 + 0.34). This means that the size of the system is less than or equal to 201 sizel (= 184 + 17) 84% of the time. Because effort is directly proportional to size, and cost is directly proportional to effort, you can use this information to set a “safe” bid price. The project suffers a cost overrun only if the actual size turns out to be above the value used for the bid. (This ignores other possible causes of cost overruns.) A good manager can typically influence the project cost by 20%; it should be feasible to complete the project for the cost estimated using the size of 201 sizel. (This is obviously overly simplified, but conveys the concept.) Figure 4‑11 Probability distribution for total size

Source: (Stutzke R. D., 2005)

Table 4-9 shows how to choose a safe value given any desired probability. For example, choose a size of (μ + 1.3*σ) if you want to have a 90% probability of not exceeding that size value. Because the normal curve falls off rapidly, a small increase in bid price “buys” a lot of confidence. For example if σ/μ = 0.1, choosing a size of (μ + σ) increases the size (and the cost) by only 10%, but increases the probability by 34%.

Table 4‑9 Choosing a Safe Value

 Size Value Probability That the Size Will Not Exceed the Size Value μ 50% μ + 0.5σ 59% μ + 0.8σ 79% μ + 1.0σ 84% μ + 1.3σ 90%

(Stutzke R. D., 2005)

The most questionable part of this method is the distribution assumption. The distribution may not be always normal distribution. Triangular distribution may be assumed to be more realistic in some projects. It also depends on the estimations of team members which may differ according to experience.