Some Results Concerning the Estimation of Beta Distribution Parameters in PERT

The traditional method of estimating mean activity-time in PERT analysis, as a weighted average of the pessimistic, most likely and optimistic activity-times, is examined. Based on properties of the beta distribution, it is shown that there are theoretical grounds for using this estimate for a specified range of modal activity times. Finally, a simple refinement of the estimate is suggested when the modal value lies outside this range.     

A New Approach to the Activity-time Distribution in PERT

We consider the problem of computing the density function of the completion time of PERT stochastic networks. An activity-time distribution is suggested which is stable to maximization and close to the beta distribution. Thus classical network-reducing methods, e.g. Martin's algorithm, can be essentially simplified.     

Mixture densities for project management activity times: A robust approach to PERT

PERT is a widely utilized framework for project management. However, as a result of underlying assumptions about the activity times, the PERT formulas prescribe a light-tailed distribution with a constant variance conditional on the range. Given the pervasiveness of heavy-tailed phenomena in business contexts as well as inherently differing levels of uncertainty about different activities, there is a need for a more flexible distribution which allows for varying amounts of dispersion and greater likelihoods of more extreme tail-area events. In particular, we argue that the tail-area decay of an activity time distribution is a key factor which has been insufficiently considered previously. We provide a distribution which permits varying amounts of dispersion and greater likelihoods of more extreme tail-area events that is straightforward to implement with expert judgments. Moreover, the distribution can be integrated into the PERT framework such that the classic PERT results represent an important special case of the method presented here.     

A dependent project evaluation and review technique: A Bayesian network approach

The Program Evaluation and Review Technique (PERT) dates back to 1959. This method evaluates the uncertainty distribution of a project’s completion time given the uncertain completion times of the activities/tasks comprised within it. Each activity’s uncertainty was defined originally by a unique two parameter beta PERT distribution satisfying what is known to be the PERT mean and PERT variance. In this paper, a three-parameter PERT family of bounded distributions is introduced satisfying that same mean and variance, generalizing the beta PERT distribution. Their additional flexibility allows for the modeling of statistical dependence in a continuous Bayesian network, generalizing in turn the traditional PERT procedure where statistical independence is assumed among beta PERT activity durations. Through currently available Bayesian network software and the construction of that PERT family herein, the coherent monitoring of remaining project completion time uncertainty given partial completion of a project may become more accessible to PERT analysts. An illustrative example demonstrating the benefit of monitoring of remaining project completion time uncertainty as activities complete in that Bayesian fashion shall be presented, including expressions and algorithms for the specification of the three prior parameters for each activity in the project network to adhere to classical the PERT mean and PERT variance and a degree of statistical dependence between them.     

Transportation systems and residential location

An optimization model for the spatial allocation of residents in an urban region is presented. The model allocates an exogenously given population on the basis of travel time (to workplaces) and population density criteria and subject to local and subregional capacity constraints. A way of calibrating the ‘value’ parameters of the model on a short-term plan proposal is outlined. Some applications of the model to planning problems of the Stockholm region are reported. These include sensitivity analyses of the population distribution of an adopted plan with respect to its exogenous assumptions (particularly those describing the transportation system). When evaluating investments with long life times, there is a strong need for long-term land use scenarios. Some long-range applications in the field of traffic and energy planning are reported. They can be seen as ‘prolongations’ of the adopted regional plan based on the preferences ‘revealed’ by the calibration procedure. Further extensions of the model to cover cost and energy considerations of urban developments are also outlined.     

航空复杂产品项目活动时间混合分布估算模型的改进

航空复杂产品开发项目具有复杂性、随机性、多目标性的特点,并且产品本身小批量、多品种的生产模式使得无法大量积累历史数据。因此,航空复杂产品项目活动时间存在高度的不确定性。如何能够更加合理、准确的描述其项目活动时间,对优化航空复杂产品的研制过程、缩短研制周期以及降低研制成本具有重大的意义。本文在Hahn基于PERT所建立的Beta Rectangular混合分布模型的基础上,保留其期望值的表达式,对其方差表达式进行改进。按照Malcolm、José等人的思想,将最可能值m与Beta分布的众数相对应,并且考虑其对方差的影响。仅保留PERT的一个假设,使方差的推导更多地依照于Beta分布而不是较多的近似。仿真结果表明,改进后的混合分布模型不但更具柔性,而且可以更加科学准确地描述航空复杂产品调度中高度不确定的任务持续时间状况。     

A New Distribution Function for Risk Analysis

This paper presents an alternative to the beta continuous probability distribution for risk analysis. Particular attention has been given to two major applications of distributions, namely project management risk and critical path analysis (PERT). In conjunction with the beta, the triangular and normal distributions are frequently employed in order to give sufficient robustness to risk analysis. The beta distribution, as used in PERT, has a major theoretical implementation flaw. The new distribution was developed to give a possible alternative method of assessing risk. It is shown that the requirement to estimate the most pessimistic variate may be replaced by the probability to exceed the mode. Proposals for other simplifications in risk analysis are discussed. Practical means to validate the most appropriate distributions for risk analysis are outlined, and a cost-data case study is included.     

Applications of decision networks

In the last three decades, the ‘Principle of Optimality’ has been applied to a number of problems of multi-stage character. In this paper, we wish to review some of the work done by different authors using the principle.The current emphasis is on Decision Networks, which incorporates the basic principle and yet reveals the graphical structure of the problem similar to PERT/CPM charts. Actual applications of this technique are discussed in detail. It is expected that many new and real-life problems will be amenable to this technique.     

Motor premium rating

This paper outlines a very simple structure for the practical analysis of motor premiums from a UK point of view. It indicates that it is possible to model different types of claims cost. These results are combined together with some simple economic assumptions, to arrive at premiums. Then the paper develops a ‘points system’ which is similar to a number of premium systems operated by UK Insurance Companies. This ‘points system’ is used to compare different sets of assumptions. Finally, an analysis of surplus is described with an example.     

Measuring ecological efficiency with data envelopment analysis (DEA)

The measurement of ecological efficiency provides some important information for the companies’ environmental management. Ecological efficiency is usually measured by comparing environmental performance indicators. Data envelopment analysis (DEA) shows a high potential to support such comparisons, as no explicit weights are needed to aggregate the indicators. In general, DEA assumes that inputs and outputs are ‘goods’, but from an ecological perspective also ‘bads’ have to be considered. In the literature, ‘bads’ are treated in different and sometimes arbitrarily chosen ways. This article aims at the systematic derivation of ecologically extended DEA models. Starting from the assumptions of DEA in production theory and activity analysis, a generalisation of basic DEA models is derived by incorporating a multi-dimensional value function f. Extended preference structures can be considered by different specifications of f, e.g. specifications for ecologically motivated applications of DEA.     

Kinship systems and inverse semigroups

Inverse semigroups are introduced as a model for kinship structures. It is shown that semigroups are implied by the traditional anthropological law of uniform descent, and that inverses are suggested by the principle of reciprocity. A separation into a ‘surface’ and a ‘deep’ level of analysis was induced by viewing a kinship system as a partition on an underlying inverse semigroup. Some Omaha‐type kinship systems are decomposed into direct and semidirect products. Additional assumptions are made to deiive a theoretical similarity measure, which is then compared with a judged similarity task on Tenejapan Indians. The results of this preliminary experiment seem to be encouraging.     

A new probabilistic approach to the path criticality in stochastic PERT

The notion of critical path is a key issue in the temporal analysis of project scheduling in deterministic setting. The very essence of the CPM consists in identifying the critical path, i.e., the longest path in a project network, because this path conveys information on how long it should take to complete the project to the project manager. The problem how can a stochastic counterpart of the deterministic critical path be defined is an important question in stochastic PERT. However, in the literature of stochastic PERT this question has so far almost been ignored, and the research into the random nature of a project duration has mainly been concentrated on the completion time in stochastic PERT in which any concrete special path is not specified. In the present paper we attempt to take first steps to fill this gap. We first developed a probabilistic background theory for univariate and bivariate marginal distributions of path durations of stochastic PERT whose joint path durations are modelled by multivariate normal distribution. Then, a new probabilistic approach to the comparison of path durations is introduced, and based on this comparison we define the concept of probabilistically critical path as a stochastic counterpart of the deterministic critical path. Also, an illustrative simple example of PCP and numerical results on the established probability bounds are presented.     

Polar and singular value decomposition of 3×3 magic squares

In the literature there have been reported a number of studies of probabilities involved in Monopoly, the worldwide popular game designed by Charles B. Darrow in the 1930s. Simple assumptions often lead to inaccurate results. In this paper, having distinguished between ‘In Jail’ and ‘Just Visiting’, both theoretical computations and computer simulations are conducted to determine the probability of landing on ‘JAIL’. Theoretical and simulation results are consistent. The sum of the probabilities of landing on ‘In Jail’ and ‘Just Visiting’ is found to be 6.40%, both in theory and in simulation, compared with Ian Stewart's 5.89%. The long-term probabilities of landing on the other 39 spaces are computed and it is concluded that the best properties to buy are the orange ones.     

Critical perspectives of the ‘new’ difference equation solution method of Rivera-Figueroa and Rivera-Rebolledo

Recently, claims of a ‘new and straightforward’ method of solution to second-order linear difference equations have appeared in the mathematics education literature from Rivera-Figueroa and Rivera-Rebolledo. The claim of novelty is based on an assumption that ‘since the equation is worked in its canonical form’, the method within this context must be new. In addition, the assertion of straightforwardness is based on the position that ‘the solution comes naturally’ through this method, rather than artificially. In this article, we subject these claims and assumptions to closer scrutiny, examination and analysis. We note that the method has been published before, and we present the method in a more succinct form. We also discuss how the method can be extended to solve difference equations with non-constant coefficients, illustrating this via a discussion of an example.     

Estimating a Project's Completion Time Distribution Using Intelligent Simulation Methods

Other researchers have indicated the theoretical problems of using the PERT techniques commonly described in elementary operational research and operations management texts. Monte Carlo simulation has often been suggested as an alternative means of analyzing PERT networks but simulating a network of real world proportions is often assumed to be prohibitively expensive. This paper measures the accuracy and cost of standard PERT and simulation methods for real world sized problems. In addition, several computational heuristics are described and tested that indicate that simulation is a viable alternative. The results indicate that intelligent simulation of PERT networks is considerably more accurate than standard PERT analysis and is definitely not cost prohibitive.     

Stability of jurisdiction structures in economies with local public goods

Individuals of different types can form groups, i.e. jurisdictions, for the purposes of collective consumption and production of local public goods by the members of the jurisdictions. Also, the utility of an individual may be affected by the composition and size of the jurisdiction of which he is a member. Jurisdiction formation is endogenous. Trade of private goods can occur within jurisdictions and within collections of jurisdictions. A stable partition of individuals is shown to exist for all sufficiently large economies. This stability depends, partially, upon the extent of ‘satisficing’ behavior or alternatively, jurisdiction formation costs, both of which can be made arbitrarily small. The major noteworthy assumption is that positive outputs cannot become virtually free in per-capita terms as the economy is replicated; this ensures that the public goods are ‘local’ rather than ‘pure’; otherwise assumptions on production sets are minimal and, in particular, convexity is not required. To obtain stability with coalition formation costs, additional assumptions are made ensuring that there is a ‘minimum efficient scale’ for coalitions.     

The Most Critical Path in a PERT Network

The classical PERT approach uses the path with the largest expected duration as the critical path to estimate the probability of completing a project by a given deadline. However, in general, such a path is not the ‘most critical’ path and does not provide the smallest estimate for the probability of completion time. This paper studies the ‘most critical path’ problem and formulates it as an optimal path problem in a deterministic network with a two-attribute fractional objective function. An exact solution approach is presented for the optimal path problem which also gives the solution to the most critical path problem. The illustrative examples as well as our computational results demonstrate that the proposed algorithm provides estimates for the probabilities of completion time that are much more accurate than those of the classical approach.     

An upper bound on the expected completion time of PERT networks

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.     

Revisiting the PERT mean and variance

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.     

The robustness of mean and variance approximations in risk analysis

This paper examines further the problem of estimating the mean and variance of a continuous random variable from estimates of three points within the distribution, typically the median or mode and two extreme fractiles. The problem arises most commonly in PERT and risk analysis where it can usually be assumed that the distribution in question is bell-shaped and positively skewed, often typified by a Beta distribution. Over the years, a number of alternative approximations have been proposed, usually as modifications to the original PERT formulae. The accuracy of a number of these approximations is investigated based not only on a Beta distribution, but also for three other commonly used bell-shaped, positively skewed distributions, namely the Gamma, Lognormal and F distributions. It is shown that a balanced weighted average of the median and the 4% fractiles provides a consistent estimator of the distribution mean across all four distributions. Furthermore, reasonably accurate estimates of the variance can also be obtained by treating the three fractiles as defining an equivalent discrete distribution with the same probability weight as in the formula for the mean.