A Review of Activity Time Distributions in Risk Analysis <s> Activity Time Distributions in Literature <s> It is assumed that activity durations in a PERT network are independent and normal random variables. A simple method of obtaining lower and upper bounds for the expected project completion time is described. The tightness of the bounds is examined for some numerical examples. A problem of applying the method in the case of non-normal distributions of activity times is also discussed. <s> BIB001 </s> A Review of Activity Time Distributions in Risk Analysis <s> Activity Time Distributions in Literature <s> We consider the PERT model of a project composed of activities whose durations are random variables with known distributions. For the situations in which the activity durations are completely independent, we present a new method for obtaining a probability distribution function that bounds the exact probability distribution of the project completion time from below. The bounding distribution can be used to obtain an upper bound on the mean completion time of the project. We also prove and illustrate that this bounding distribution is better tighter than any of the existing lower bounds, implying that the corresponding upper bound on the mean completion time is tighter than any of the existing upper bounds. <s> BIB002 </s> A Review of Activity Time Distributions in Risk Analysis <s> Activity Time Distributions in Literature <s> Abstract In this paper the problem of determining an upper bound on the expected project completion time, described by the PERT network, is considered. It is assumed that activity durations are independent random variables with given means. The exact forms of probability distributions do not have to be known; however, their cumulative distribution functions are expected to belong to the so-called NBUE class. Very simple algorithms for deriving this bound are presented. The computations can even be performed manually for more involved networks. Our approach producing a pessimistic evaluation of the expected value of the project duration, extends considerably the information obtained through the use of the classical PERT that always underestimates this value. The results are illustrated by a simple example, and errors of approximations are discussed. <s> BIB003 </s> A Review of Activity Time Distributions in Risk Analysis <s> Activity Time Distributions in Literature <s> This paper investigates pert networks with independent and exponentially distributed activity durations. We model such networks as finite-state, absorbing, continuous-time Markov chains with upper triangular generator matrices. The state space is related to the network structure. We present simple and computationally stable algorithms to evaluate the usual performance criteria: the distribution and moments of project completion time, the probability that a given path is critical, and other related performance measures. In addition, we algorithmically analyze conditional performance measures-for example, project completion time, given a critical path-and present computational results. We then study extensions both to resource-constrained pert networks and to a special class of nonexponential pert networks. <s> BIB004 </s> A Review of Activity Time Distributions in Risk Analysis <s> Activity Time Distributions in Literature <s> A simplified version of the program evaluation and review technique (PERT) for project planning is developed and tested. The simplification is to reduce the number of estimates required for activity durations from three, as in conventional PERT, to two. This is accomplished by applying the normal distribution, rather than the beta, to an activity duration. The two required duration estimates are the “most likely” and the “pessimistic.” These modifications reduce the level of effort needed to apply PERT. Simplified PERT durations are subject to errors of greater than 10% when the skewness of the actual distribution is greater than 0.28 or less than −0.48. In analyzing 12 project networks, though, the simplified PERT produced values similar to those of conventional PERT for activity and project durations and variances and project duration probabilities. Hence, when activity duration distributions are not highly skewed, results similar to those of conventional PERT can be obtained using the simpler technique... <s> BIB005 </s> A Review of Activity Time Distributions in Risk Analysis <s> Activity Time Distributions in Literature <s> This paper presents an approach for using right-truncated ex- ponentially distributed random variables to model activity times in sto- chastic activity networks. The advantages of using the right-truncated exponential distribution are discussed. The moments of a project comple- tion time using the proposed distribution are derived and compared with other estimated moments in literature. <s> BIB006 </s> A Review of Activity Time Distributions in Risk Analysis <s> Activity Time Distributions in Literature <s> The problem of determining the moment of the project completion time (MPCT) with stochastic activity durations is addressed. In the direct approach the MPCT is obtained after analyzing all possible network paths which is NP-hard problem. A recursive method for determining MPCT by proceeding iteratively over the nodes is the computationally least demanding and therefore the most efficient approach. Unfortunately, the moment methods which are available in literature ignore some aspects. In this paper, an adjustment of the moment completion time is discussed. Two illustrative examples are presented. <s> BIB007 </s> A Review of Activity Time Distributions in Risk Analysis <s> Activity Time Distributions in Literature <s> When managing projects with considerable uncertainty, such as those arising in construction, defense, and new product development, it is customary for a manufacturer (project manager) to offer contracts under which each supplier (contractor) receives a prespecified payment when she completes her task. However, there are recent cases in which the manufacturer imposes “delayed payment” contracts under which each supplier is paid only when all suppliers have completed their tasks. By considering a model of one manufacturer and n ≥ 2 identical and independent suppliers with exponential completion times, we analyze the impact of both a delayed payment regime and a no-delayed-payment regime on each supplier's effort level and on the manufacturer's net profit in equilibrium. When the suppliers' work rates are unadjustable, we conjecture that the manufacturer is actually worse off under the delayed payment regime. However, when the suppliers' work rates are adjustable, we obtain a different result: the delayed payment regime is more profitable for the manufacturer either when the project revenue is sufficiently small or when the number of suppliers is sufficiently large. <s> BIB008 </s> A Review of Activity Time Distributions in Risk Analysis <s> Activity Time Distributions in Literature <s> Difficulties with the interpretation of the parameters of the beta distribution let Malcolm et al. (1959) to suggest in the Program Evaluation and Review Technique (PERT) their by now classical expressions for the mean and variance for activity completion for practical applications. In this note, we shall provide an alternative for the PERT variance expression addressing a concern raised by Hahn (2008) regarding the constant PERT variance assumption given the range for an activity's duration, while retaining the original PERT mean expression. Moreover, our approach ensures that an activity's elicited most likely value aligns with the beta distribution's mode. While this was the original intent of Malcolm et al. (1959), their method of selecting beta parameters via the PERT mean and variance is not consistent in this manner. <s> BIB009
1) The Beta Distribution The originators of classical PERT  assumed that project activity time follows the generalized beta distribution with probability density function where α and β are the shape parameters, ( ) is the gamma function. The mean, variance and skewness are respectively given as BIB009 . This 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  worked on the improvement of the classical PERT estimates based on only two subjective estimates the pessimistic 1) and optimistic 2) times. He posited that analysis of many project networks with lengthy periods reveals that the most likely activity time is practically useless. He pointed out that its relative location in time interval [ ]  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  Also, substituting 1 p + and 1 q + for p and q respectively they obtained the 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 ( 2 σ ) if the probability density function is given as Kamburowski BIB001 assumed that the activity durations of PERT network are independent and normally distributed random variables. He obtained a lower and upper bounds for the expected project completion time using a simple recursive algorithm. The tightness of the bounds was examined for some numerical examples. It was apparent that the lower bounds are tighter than the estimates of the classical PERT procedure when empirical activity distribution is symmetric but otherwise when the distribution is asymmetric. Kamburowski's method BIB001 followed after Dodin BIB002 . Sculli  proposed an approximation for the completion time mean and variance of PERT networks. His method assumed that activity durations are normally and independently distributed. He further assumed that various paths of the network are independent, and that the network can be transformed into the type where only maximum of two activities terminate on the same event, such that the problem of finding the distribution of ( ) are independent normal random variable with mean i µ and common variance 2 σ . He demonstrated that his method which adopts path independence produced better results than the classical PERT method which assumes complete dependence of paths. Drezner and Anklesaria  also developed a method for solving PERT networks as a multivariate problem taking into consideration path correlation. They assumed that each path duration is the sum of activities on the path, and then defined ; 1, 2, , resulting to an m-dimensional integral problem. Their method was not popular because of much computation time required for an approximate solution to be obtained even for small project networks. Cottrell BIB005 developed a simplified version of PERT using normally distributed activity times. The simplification was obtained by reducing the number of estimates required for activity durations from three, as in classical PERT, to two (the most likely-m, and the pessimistic-b times) which were subjectively chosen. In such case, the most likely time (m) coincided with the mean, and the variance was obtained using ( ) 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  also considered a doubly truncated normal distribution for the activity time distribution in PERT via a maximum entropy approach. 3) The Exponential Distribution The exponential distribution has been used to describe activity times. Magott and Skudlarski  , Abdelkader and Mouhamed  used the exponential distribution as a representation of activity duration in Stochastic activity Networks Abdelkader BIB007 later presented an adjustment to the recursive method of determining the moments of the project completion times in SANs when activity times are exponentially distributed. But one of the criticisms of using the exponential distribution is that it assumes a constant probability of completion in the next time period, irrespective of the elapsed activity duration. Hence, Abd-el-Kader BIB006 used the truncated exponential distribution as the activity time distribution. He equally adopted Stochastic Activity Networks (SANs) approach to obtain the moments of the project completion time. His effort yielded an improvement on the estimates obtained using the untruncated exponential distribution. Cinicioglu & Shenoy  described how a stochastic PERT network can be transformed into a mixture of truncated exponentials Bayesian network. They adopted the Lauretsen-Jensen algorithm for solving mixtures of Guassian (MoG) hybrid bayesian networks and further approximated a PERT Bayesian network by MoG Bayes net. Their method suffered a setback during arc reversal in complex activity networks. Azaron and Modarres  transformed a dynamic PERT network with exponential activity duration in into stochastic network and then obtained the project completion time by constructing Continuous Time Markov Chain (CTMC). Other works on exponentially distributed activity times could be found in Kamburowski BIB003 , Kulkarni and Adlakha BIB004 and Kwon, et al. BIB008 .