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Project Evaluation and Review Technique (PERT) alongside recent modifications is a popular and useful tool in project risk analysis. Over the past seven decades, there have been some modifications in PERT owing to the shift from beta distributed activity times to other activity time distributions. This paper presents a review of activity time distributions in risk analysis as found in literature up to date.

Project Evaluation and Review Technique (PERT) is widely used by project managers and practitioners as the probabilistic form of the Critical Path Method (CPM). The PERT method is not only useful for the estimation of project completion times but it is also workable and cost-effective for management of projects [

Although it is well accepted that the classical PERT gives useful estimates, its assumption introduces some potential sources of bias which results in the underestimation of project completion time [

The importance of probability distributions in PERT cannot be overemphasized as both the simulation and analytical approaches assume probability distributions for activity durations a priori or a posteriori [

In this paper, we present a review of the various activity duration distributions that have been used for the analysis of project networks. Some modified versions of PERT-Beta approach are also presented. We further highlight the various methods adopted for parameter estimation based on these distributions.

1) The Beta Distribution

The originators of classical PERT [

f ( x ) = Γ ( α + β ) Γ ( α ) Γ ( β ) ( x − a ) α − 1 ( b − x ) β − 1 ( b − a ) α + β − 1 ; a < x < b , α , β > 0

where α and β are the shape parameters, Γ ( . ) is the gamma function. The mean, variance and skewness are respectively given as

μ x = a + ( b − a ) α α + β , σ x 2 = ( b − a ) 2 α β ( α + β ) 2 ( α + β + 1 )

and

γ 1 = 2 ( β − α ) α + β + 1 ( α + β + 2 ) α β

In classical PERT, the mean and variance were estimated to be μ ^ x = a + 4 m + b 6 and σ ^ x 2 = ( b − a ) 2 36 . A study by Farnum and Stanton [

of the beta distribution in classical PERT is appropriate within some range of modal values, namely, a + 0.13 ( b − a ) < m < b − 0.13 ( b − a ) . This means that the estimate performs poorly outside this interval. This can either happen when the most likely estimate, m, is chosen to be very close to the two extreme values, a and b, (less than 13% of the range from either a or b). In other words the classical PERT estimate fails when activity time distributions are heavily tailed. Moreover, previous works reveal that the classical PERT assumptions of the mean and variance restrict us to only three members of the beta family, namely, 1) α = β = 4 ; 2) α = 3 − 2 , β = 3 + 2 ; 3) α = 3 + 2 , β = 3 − 2 . In which

case the skewness will be 0, 1 2 and − 1 2 respectively [

restriction led to various modifications on the classical PERT to accommodate more members of the beta family. We will discuss some of these modifications. Most of these modifications are based on the adjustments of the parameters of beta distribution.

Gollenko-Ginzburg [

m = 2 a + b 3 . Given the density function f ( x ) = Γ ( p + q + 2 ) Γ ( p + 1 ) Γ ( q + 1 ) x p ( 1 − x ) p ( m x − 1 )

( m x = mode of x) which was obtained after a re-parametisation of the standard beta distribution, with additional assumption that p + q ≅ Z (constant). The

following results μ y = 2 a + 9 m + 2 b 13 and

σ y 2 = ( b − a ) 2 1268 [ 22 + 81 ( m − a b − a ) − 81 ( m − a b − a ) 2 ] were obtained for the estimation

of the mean and variance of activity distribution. He showed that these formulae provide better results as compared to the classical pert formulae when the estimated mode is located in the tails of the distribution. These formulae were further reduced to μ y = 0.2 ( 3 a + 2 b ) and σ y 2 = 0.04 ( b − a ) 2 on the basis of the

earlier assumption of the mode, ≅ 2 a + b 3 . A similar modification was carried

out by Shankar and Sireesha [

f ( x ) = Γ ( p + q + 2 ) Γ ( p + 1 ) Γ ( q + 1 ) x p ( 1 − x ) p ( m x − 1 ) with the relation p + q ≅ K (constant). Also, substituting p + 1 and q + 1 for p and q respectively they obtained the results μ x = 17 m x + 5 27 and σ x 2 = ( 17 m x + 10 ) ( 27 − 17 m x ) 2300 which give

μ x = 5 a + 17 m + 5 b 27 and σ x 2 = ( 17 m − 27 a + 10 ) ( 27 b − 10 a − 17 m ) 2300 for the ge-

neral beta distribution. Their method further created allowance for the accommodation of some events in the tail of the distribution. Trout [

2) The Normal Distribution

The proponents of the normal activity time distribution posit that activity times can as well be normally distributed regardless of the popular opinion of the right skewed activity times. A random variable X is said to be normally distributed with mean ( μ ) and variance ( σ 2 ) if the probability density function is

given as f ( x ) = 1 σ 2 2 π e ( x − μ ) 2 2 σ 2 ; − ∞ < x < ∞ . Its coefficient of skewness is zero.

Kamburowski [

and the variance was obtained using σ 90 2 ( T ) = ( b − m 1.645 ) 2 . Although his method

seemed to reduce the effort needed to apply PERT, it was subject to errors greater than 10% when the skewness of the actual distribution is greater than 0.28 or less than −0.48. Kotiah and Wallace [

3) The Exponential Distribution

The exponential distribution has been used to describe activity times. Magott and Skudlarski [

mean variance, and skewness are μ x = 1 λ , σ x 2 = 1 λ 2 , and γ 1 = 2 respectively.

Abdelkader [

4) The Weibull Distribution

The Weibull distributed activity time was considered by Abd-El-Kader [

has mean, variance and skewness given as μ x = θ Γ ( 1 + 1 β ) ,

σ x 2 = θ 2 [ Γ ( 1 + 2 β ) − ( Γ ( 1 + 1 β ) ) 2 ] and γ 1 = Γ ( 1 + 3 β ) θ 3 − 3 μ x σ x 2 − μ x 3 respectively.

5) The Lognormal Distribution

A random variable X is lognormal if the probability density function is given as

f ( x ) = 1 x 2 π σ e − ( I n x − μ ) 2 2 σ 2 ; x > 0

Its mean, variance, skewness are μ x = e μ + σ 2 2 , σ x 2 = ( e σ 2 − 1 ) e 2 μ + σ 2 , and γ 1 = ( e σ 2 − 2 ) e σ 2 − 1 respectively. Mohan et al. [

approximation of activity duration in PERT using two time estimates. His method handled the heavy tailed property of the activity time distribution which is deficient when normal activity time is assumed and also reduced the parameters from three (a-Optimistic, m-Most likely, and b Pessimistic) to two (a-Optimis- tic, and m-Most likely) or (m-Most likely, and b-Pessimistic). It was demonstrated with examples that their methods are better than the normal approximation when the underlying activity distribution is skewed to the right and better than the classical PERT method only when the activity distribution is heavily right skewed. Trietsch, et al. [

6) The Triangular Distribution

The triangular distribution has also been suggested as a priori distribution for activity times. Mac Crimmon and Ryavec [

g ( x ) = { 2 b − a x − a m − a ; a ≤ x ≤ m 2 b − a b − x b − m ; m ≤ x ≤ b

where m stands for the mode and the interval [ a , b ] determines the range of the random variable X. The mean, variance and skewness are given as

μ x = a + m + b 3 , σ x 2 = a 2 + b 2 + m 2 − a b − a m − b m 18

and

γ 1 = 2 ( a + b − 2 m ) ( 2 a − b − m ) ( a − 2 b + m ) 5 ( a 2 + b 2 + m 2 − a b − a m − b m )

The a, m and b could be obtained intuitively as in the case of the classical PERT. Johnson [

7) The Uniform Distribution

MacCrimmon and Rayvec [

with probability density function given as f ( x ) = 1 b − a ; a < x < b with mean, variance, and skewness μ x = a + b 2 , σ x 2 = ( b − a ) 2 12 and γ 1 = 0 respectively. Re-

cently, Abdelkader and Al-Ohali [

8) The Erlang Distribution

Bendell, et al. [

density f ( x ) = λ k Γ ( k ) e − λ x x k − 1 ; x ≥ 0 ; λ , k > 0 .

Its mean, variance, and skewness are μ x = k λ , σ x 2 = k λ 2 , and γ 1 = 2 k re-

spectively. They obtained the first four central moments of the Max ( X 1 , X 2 ) , where X 1 and X 2 are independent random variables, and further demonstrated the accuracy of their method in many practical scenario. Their method formed the basis upon which multi-modal activity time distributions could be used. Abdelkader [^{th} moments of the Max ( X 1 , X 2 , ⋯ , X n ) and the cumulative distribution function of the sum of n independent random variables.

9) The Gamma Distribution

Lootsma [

f ( x ) = { λ α Γ ( α ) ( x − a ) α − 1 exp { − λ ( x − a ) } ; x ≥ a 0 ; x < a

where α is the shape parameter and λ is the scale parameter of the gamma

distribution. Its mean, variance and skewness are μ x = α λ + a and σ x 2 = α λ 2 and γ 1 = 2 α respectively. The estimates of the mean and variance were given as μ ^ x = b − 5 m 6 and σ ^ x 2 = ( μ ^ x − m ) ( μ ^ x − a ) , based on intuitive time estimates

(the optimistic (a), most likely (m) and pessimistic (b) times) from the practitioner. Abdelkader [

10) The Compound Poisson distribution

Parks and Ramsing [

11) The Beta Rectangular Distribution

A mixture density, beta-rectangular distribution was introduced by Hahn [

p ( y , α , β , θ , a , b ) = θ Γ ( α + β ) ( y − a ) α − 1 ( b − y ) β − 1 Γ ( α ) Γ ( β ) ( b − a ) α + β − 1 + 1 − θ b − a

where θ is the mixing parameter on interval 0 ≤ θ ≤ 1 . The mean and variance of the mixture density are

μ y = a + ( b − a ) ( θ α k + 1 − θ 2 )

and

σ y 2 = ( a + b ) 2 ( θ α ( α + 1 ) k ( k + 1 ) + 1 − θ 3 − ( k + θ ( α − β ) ) 2 4 k 2 )

The mean and variance were approximated as

μ ^ x = θ ( a + 4 m + b ) + 3 ( 1 − θ ) ( a + b ) 6

σ ^ x 2 = 1 36 [ θ ( ( a + 4 m + b ) 2 + ( b − a ) 2 ) + 12 ( 1 − θ ) ( a 2 + a b + b 2 ) − ( θ ( a + 4 m + b ) + 3 ( 1 − θ ) ( a + b ) ) 2 ]

respectively. His method, in comparison with the classical PERT method, accommodated greater likelihood of more extreme tail- area events that seemed straight forward to implement with experts judgment. However, in addition to the three intuitive parameters ( a , m , b ) of classical PERT, his method introduced the fourth parameter θ which should also be subjectively chosen by project managers. Yakhchali [

12) Tilted Beta Distribution

Hahn and Martín [

p ( x / v , α , β , θ ) = { ( 1 − θ ) [ 2 v − 2 ( 2 v − 1 ) x ] + θ [ Γ ( α + β ) Γ ( α ) Γ ( β ) x α − 1 ( 1 − x ) β − 1 ] ; 0 ≤ x ≤ 1 0 ; otherwise

where θ ∈ [ 0 , 1 ] and θ ∈ [ 0 , 1 ] . The mean and variance of the tilted beta distribution are

( 1 − θ ) 2 − v 3 + θ α α + β

and

( ( 1 − θ ) 3 − 2 v 6 − θ α ( α − 1 ) ( α + β ) ( α + β + 1 ) ) − ( ( 1 − θ ) 2 − v 3 + θ α α + β ) 2

( ( 1 − θ ) 3 − 2 v 6 − θ α ( α − 1 ) ( α + β ) ( α + β + 1 ) ) − ( ( 1 − θ ) 2 − v 3 + θ α α + β ) 2

respectively. The distribution is a mixture of the tilting distribution [

we obtain the tilted distribution, if θ = 1 2 either the beta distribution, uniform

distribution, or beta rectangular distribution is obtained depending on the value of v. The parameters of the distribution where elicited as follows: Given the beta distribution with k = α + β = 6 ; α ≠ β and noting that in this case the mean

and mode are α k and α − 1 k − 2 respectively. Solving some simultaneous equa-

tions α and β where recomputed as 4 m + 1 and 5 + 4 m for the standardized beta. To elicit v, it was assumed that there exists a linear increase or decrease in the probability density across time in accordance with the shape of the tilting distribution. Hence, the expert is requested to estimate the probability of the event of activity completion in day j (say) denoted by P ( A j ) as well as the probability of the event of completion in day j + 1 , denoted by P ( A j + 1 ) . Equat-

ing the rate of change denoted by r = P ( A j ) − P ( A j + 1 ) 1 / ( b − a ) (a = optimistic time, b = pessimistic time) to the slope of the tilted density function, − 2 ( 2 v − 1 ) and solving yields v = 2 − r 4 . The mixing parameter θ was elicited as a judgmental

estimate as in Hahn [

13) Burr XII Distribution

The Burr type 12 distribution [

f ( x ) = c k x α c c − 1 ( 1 + ( x α ) c ) − ( k + 1 ) ; x ≥ 0 , c > 0 , α > 0 , k > 0

The cumulative distribution function is F ( x ) = 1 − ( 1 + ( x α ) c ) − k . The r^{th} moment about the origin is given as

E ( X r ) = k α r Γ ( k − r c ) Γ ( r c + 1 ) Γ ( k + 1 ) ; c k > r

We have presented an up-to-date review of the activity time distributions used in PERT with highlights of various methods adopted for parameter estimation. From the review, three estimation approaches are outstanding, namely, Analytical Approximation, Monte Carlo Simulation and SANs, see

Monte Carlo Simulation has proved to be a versatile technique with regards to the choice of distributional forms. Apart from the exact technique, the simulation technique has the capacity to produce more efficient results PMBOK [

Probability Distribution | Method of Estimation |
---|---|

Beta | Analytical approximation. Monte Carlo Simulation. |

Normal | Analytical approximation. Analytical bounding. Exact analysis. |

Exponential | SANs |

Weibull | Analytical approximation. SANs |

Lognormal | Analytical approximation |

Triangular | Analytical approximation. Monte Carlo Simulation. |

Uniform | Analytical approximation. SANs. Monte Carlo Simulation. |

Erlang | SANs. |

Gamma | Analytical approximation. SANs. |

Compound Poisson | Analytical approximation |

Beta-Rectangular | Analytical approximation. |

Tilted-Beta | Analytical approximation. |

Burr XII | Monte Carlo Simulation |

A basic advantage of the Simulation approach is that it allows the use of any activity time distribution. In short, different distributions can be used on different activities of the same project. It was observed that the choice of most of the activity time distributions was based on flexibility and convenience, with no clear empirical evidences, as earlier noted by Trietsch et al. [

The importance of appropriate choice of activity time distribution cannot be overemphasized, irrespective of the method adopted to estimate the parameters of project network. Hence, we suggest that practitioners, apart from using theoretical information, should endeavor to make their choices of activity duration distributions based on particular empirical evidences and not just on simplicity. Developers of project management software should also incorporate many probability distributions as much as possible to enable users’ flexibility of choice. The information provided in this research can be used to extend the study by Hajdu and Bokor [

Udoumoh, E.F. and Ebong, D.W. (2017) A Review of Activity Time Distributions in Risk Analysis. American Journal of Operations Research, 7, 356-371. https://doi.org/10.4236/ajor.2017.76027