A Review of Activity Time Distributions in Risk Analysis <s> Introduction <s> A vehicle for traversing rough terrain, such as is commonly found in open mining operations and highway construction work. The vehicle is characterized by a movable frame structure. The frame includes a pair of elongated side members which are joined in a spaced apart relationship near the rear of the frame members by a first crossmember which is provided with a pivotal joint to permit the side members to rotate or pivot relative to each other. The front ends of the side members are connected by second and third crossmembers, each of which is pivotally connected at both ends to the side members to provide lateral restraint to the side members while the vehicle is traversing rough terrain. The second and third crossmembers are vertically displaced to prevent relative rotation of the side members about their longitudinal axes while permitting up and down movement of the forward ends of the side members as the side members rotate about the pivot point in the first crossmember. <s> BIB001 </s> A Review of Activity Time Distributions in Risk Analysis <s> Introduction <s> This paper presents the results of a mathematical analysis of the standard assumptions used in PERT calculations. The objectives of this analysis were four-fold: 1 to pull together the mathematical aspects of the PERT model, 2 to suggest relevant analytic techniques, 3 to obtain an indication of the magnitude and direction of errors introduced by the assumptions, and 4 to suggest possible modifications and improvements in the model. The first part of the paper analyzes those assumptions relevant to the individual activities. Three possible sources of error are considered here: 1 the beta distribution assumption, 2 the standard deviation assumption and the approximation formula for the mean, and 3 the imprecise time estimates. The second part of the paper deals with the PERT network as a whole and analyzes the calculations underlying the project mean, standard deviation, and probability statements. The concept of relative criticalness is explored for the PERT stochastic model. Techniques for network reduction are outlined. <s> BIB002 </s> A Review of Activity Time Distributions in Risk Analysis <s> Introduction <s> Two sources of PERT completion time bias are well documented in the literature: near critical paths turning critical during execution and misspecified activity time probability models. Although simulation is clearly the most appropriate method for assessing project duration, most introductory discussions touch on these issues and move quickly to standard approximations, implying that PERT offers useful, if only approximate, project duration estimates. This paper uses simulation to illustrate the nature and extent of PERT approximation errors in simple examples from two excellent texts. The examples raise serious questions about the utility of PERT project duration estimates and suggest opportunities for improvement in introductory PERT instruction. <s> BIB003 </s> A Review of Activity Time Distributions in Risk Analysis <s> Introduction <s> Recent years have seen an increase in the development of robust approaches for stochastic project management methodologies such as PERT (Program Evaluation and Review Technique). These robust approaches allow for elevated likelihoods of outlying events, thereby widening interval estimates of project completion times. However, little attention has been paid to the fact that outlying events and/or expert judgments may be asymmetric. We propose the tilted beta distribution which permits both elevated likelihoods of outlying events as well as an asymmetric representation of these events. We examine the use of the tilted beta distribution in PERT with respect to other project management distributions. <s> BIB004
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  . PERT has become of interest to management practitioners because of its simplicity, and flexibility to accommodate stochastic activity times. PERT was invented in 1958 for the POLARIS Missile Program by the Program Evaluation branch of the Special projects Office of US Navy  . Malcom in PERT. For instance, earlier works by Clark BIB001 , Kamburowski [7] supported the use of Beta distribution. Recently, Hajdu and Bokor  argued still in support of the Beta distribution. They considered the following "hypothetical" judgemental estimates: 60-optimistic, 100-most likely, and 150-pessimistic whose skewness coefficient is only 0.09826. The beta distribution defined by this coefficient of skewness is the one that approximates the normal distribution which is also supported by PERT-Beta approximation. Hence, there is no doubt about their conclusion. In spite of this, the graphical results that compared the PERT-Beta, triangular and uniform distributions still showed some obvious variations. It is possible that a set of data with much longer tail would have yielded worse results. Moreover, a comparison done with many more activity time distributions would have given more in-depth revelations. On the other hand, some researchers opine that the adoption of Beta distribution was intuitive as there was no empirical evidence for its usage. For instance, it has been demonstrated by Mac-Crimmon and Ryavec BIB002 that the error for classical PERT calculated mean and standard deviation could be up to 33% and 16% respectively. This point was buttressed by Williams BIB003 using simulation and graphical approaches to demonstrate the extent of discrepancies that exist between the beta and triangular distributions in the estimation of activity time parameters. Hahn and Martín BIB004 support the use of a more robust distribution that can accommodate outlying events. An undisputed observation in project management is that most activity time distributions are right skewed  [12]  . 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  [14]  . Consequently, researchers have suggested various activity duration distributions for the analyses of project networks. Unfortunately, most basic texts in operations research and project management present only PERT-beta distribution without any mention of other activity time distributions. 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