A linear Bayesian stochastic approximation to update project duration estimates

By relaxing the unrealistic assumption of probabilistic independence on activity durations in a project, this paper develops a hierarchical linear Bayesian estimation model. Statistical dependence is established between activity duration and the amount of resource, as well as between the amount of resource and the risk factor. Upon observation or assessment of the amount of resource required for an activity in near completion, the posterior expectation and variance of the risk factor can be directly obtained in the Bayesian scheme. Then, the expected amount of resources required for and the expected duration of upcoming activities can be predicted. We simulate an application project in which the proposed model tracks the varying critical path activities on a real time basis, and updates the expected project duration throughout the entire project. In the analysis, the proposed model improves the prediction accuracy by 38.36% compared to the basic PERT approach.     

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.     

On the Distribution of Activity Time in PERT

In PERT analysis the activity-time distribution is assumed to be a beta distribution, and the mean and variance of the activity time are estimated on the basis of the ‘pessimistic’, ‘most likely’ and ‘optimistic’ completion times, which are subjectively determined by an analyst. In this paper, on the basis of the study of the PERT assumptions, we present an improvement of these estimates. It is also shown that, by means of additional reasonable assumptions, the activity-time distribution in PERT analysis may 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.     

基于贝叶斯网络的工程项目进度完工概率分析

针对传统计划评审技术(Program Evaluation and Review Technique,PERT)在计算完工概率时假设条件的局限性(假设条件与工程实际存在偏差,导致完工概率偏大),提出了基于贝叶斯网络的施工进度完工概率分析方法.首先,分析了贝叶斯网络与进度计划网络之间的相似性,将两者结合起来构建了贝叶斯进度网络;在此基础上,综合考虑贝叶斯网络在节点取值及概率计算方面的优越性,并结合工程项目的不确定性及复杂性特点,建立了基于贝叶斯网络的施工进度完工概率分析模型.最后,将该模型应用于具体工程进行实例分析,验证了模型的可行性与有效性.研究结果表明:基于贝叶斯网络的进度完工概率模型充分考虑了工程施工中的风险因素,其结果能更客观地反映工程实际,可为工程项目决策者提供可靠的依据.     

Time-cost trade-off via optimal control theory in Markov PERT networks

We develop a new analytical model for the time-cost trade-off problem via optimal control theory in Markov PERT networks. It is assumed that the activity durations are independent random variables with generalized Erlang distributions, in which the mean duration of each activity is a non-increasing function of the amount of resource allocated to it. Then, we construct a multi-objective optimal control problem, in which the first objective is the minimization of the total direct costs of the project, in which the direct cost of each activity is a non-decreasing function of the resources allocated to it, the second objective is the minimization of the mean of project completion time and the third objective is the minimization of the variance of project completion time. Finally, two multi-objective decision techniques, viz, goal attainment and goal programming are applied to solve this multi-objective optimal control problem and obtain the optimal resources allocated to the activities or the control vector of the problem     

A new method for constructing a minimal PERT network

A project is an enterprise consisting of several activities which are to be carried out in some specific order. The activities and the order in which they need to be carried out can be represented by a PERT network. The PERT technique is a traditional, well-known approach to the expert of project management. When networks are used, it often becomes necessary to draw dummy activities. Since the computation of project completion time is proportional to the number of arcs, including dummy arcs, it is desirable to draw a network with as few dummy activities as possible.In this paper, we propose a new method for constructing, for a given project scheduling problem, a PERT network having as small as possible the number of dummy arcs by using some results on line graphs. This algorithm deals with the existence of transitive arcs. The paper contains illustrative examples, proofs of some theoretical results as well as a comparative study with a similar algorithm known in the literature. Computational results showed the superiority of our algorithm.     

The Completion Time of PERT Networks

This paper proposes an approximation for the completion time mean and variance of PERT networks in which the durations of individual activities are Normally and independently distributed. The proposed approximation involves the manipulation of distribution parameters only and hence can be computed manually, even for large networks.     

On path correlation and PERT bias

Most studies of project time estimation assume that (a) activity times are mutually independent random variables; many also assume that (b) path completion times are mutually independent. In this paper, we subject the impact of both these assumptions to close scrutiny. Using tools from multivariate analysis, we make a theoretical study of the direction of the error in the classical PERT method of estimating mean project completion time when correlation is ignored. We also investigate the effect of activity dependence on the normality of path length via simulation.     

A rule for slack allocation proportional to the duration in a PERT network

In this paper, we define a new rule for the resolution of the slack allocation problem in a PERT network. This problem exists of allocating existing extra time in some paths among the activities belonging to those paths. The allocation rule that we propose assigns extra time to the activities proportionally to their durations in such a way that no path duration exceeds the completion time of the whole project. This time allocation enables us to make a schedule for the PERT project under study. We give two characterizations of the rule and we compare it with others that have been previously defined in the literature.     

Multi-objective time–cost trade-off in dynamic PERT networks using an interactive approach

We develop a multi-objective model for the time–cost trade-off problem in a dynamic PERT network using an interactive approach. The activity durations are exponentially distributed random variables and the new projects are generated according to a renewal process and share the same facilities. Thus, these projects cannot be analyzed independently. This dynamic PERT network is represented as a network of queues, where the service times represent the durations of the corresponding activities and the arrival stream to each node follows a renewal process. At the first stage, we transform the dynamic PERT network into a proper stochastic network and then compute the project completion time distribution by constructing a continuous-time Markov chain. At the second stage, the time–cost trade-off problem is formulated as a multi-objective optimal control problem that involves four conflicting objective functions. Then, the STEM method is used to solve a discrete-time approximation of the original problem. Finally, the proposed methodology is extended to the generalized Erlang activity durations.     

Resource allocation in dynamic PERT networks with finite capacity

This article models the resource allocation problem in dynamic PERT networks with finite capacity of concurrent projects (COnstant Number of Projects In Process (CONPIP)), where activity durations are independent random variables with exponential distributions, and the new projects are generated according to a Poisson process. The system is represented as a queuing network with finite concurrent projects, where each activity of a project is performed at a devoted service station with one server located in a node of the network. For modeling dynamic PERT networks with CONPIP, we first convert the network of queues into a stochastic network. Then, by constructing a proper finite-state continuous-time Markov model, a system of differential equations is created to solve and find the completion time distribution for any particular project. Finally, we propose a multi-objective model with three conflict objectives to optimally control the resources allocated to the servers, and apply the goal attainment method to solve a discrete-time approximation of the original multi-objective problem.     

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。     

Lower bound for the mean project completion time in dynamic PERT networks

We apply the stochastic dynamic programming to obtain a lower bound for the mean project completion time in a PERT network, where the activity durations are exponentially distributed random variables. Moreover, these random variables are non-static in that the distributions themselves vary according to some randomness in society like strike or inflation. This social randomness is modelled as a function of a separate continuous-time Markov process over the time horizon. The results are verified by simulation.     

计划评审技术的改进

由于计划评审技术存在的不足,导致项目管理存在一定的偏差.对计划评审技术中的活动期望时间和方差的公式做了改进,从而提高了项目管理的精确性.还指出计划评审技术中活动期望时间和方差的计算公式,实际上是改进后公式的特殊情形.     

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.     

A lognormal approximation of activity duration in PERT using two time estimates

The success behind effective project management lies in estimating the time for individual activities. In many cases, these activity times are non-deterministic. In such situations, the conventional method (project evaluation and review technique (PERT)) obtains three time estimates, which are then used to calculate the expected time. In practice, it is often difficult to get three accurate time estimates. A recent paper suggests using just two time estimates and an approximation of the normal distribution to obtain the expected time and variance for that activity. In this paper, we propose an alternate method that uses only two bits of information: the most-likely and either the optimistic or the pessimistic time. We use a lognormal approximation and experimental results to show that our method is not only better than the normal approximation, but also better than the conventional method when the underlying activity distributions are moderately or heavily right skewed.     

A Multivariate Approach to Estimating the Completion Time for PERT Networks

Stochastic PERT is typically treated as a critical path problem, with the critical path having a univariate probability distribution. A method has been developed to treat the PERT problem as a multivariate problem taking into consideration correlation among paths, thereby providing a more accurate analysis of the network and its corresponding probabilities for completion time.     

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.     

Normally Distributed Activity Durations in PERT Networks

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.