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.     

Allocating slacks in stochastic PERT network

The SPERT problem was defined, in a game theory framework, as the fair allocation of the slack or float among the activities in a PERT network previous to the execution of the project. Previous approaches tackle with this problem imposing that the durations of the activities are deterministic. In this paper, we extend the SPERT problem into a stochastic framework defining a new solution that tries also to maintain the good performance of some other approaches that have been defined for the deterministic case. Afterward, we present a polynomial algorithm for this new solution that also could be used for the calculation of other approaches founded in the deterministic SPERT literature.     

Multi-resource allocation in stochastic project scheduling

We propose a resource allocation model for project scheduling. Our model accommodates multiple resources and decision-dependent activity durations inspired by microeconomic theory. First, we elaborate a deterministic problem formulation. In a second stage, we enhance this model to account for uncertain problem parameters. Assuming that the first and second moments of these parameters are known, the stochastic model minimises an approximation of the value-at-risk of the project makespan. As a salient feature, our approach employs a scenario-free formulation which is based on normal approximations of the activity path durations. We extend our model to situations in which the moments of the random parameters are ambiguous and describe an iterative solution procedure. Extensive numerical results are provided.     

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.     

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.     

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.     

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.     

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.     

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.     

Cost/time trade-off analysis for the critical path method: a derivation of the network flow approach

A well-known problem in critical path analysis involves normal and crash durations being provided for each activity, with corresponding costs, and requires a minimum cost schedule of durations to be determined for all possible durations of the project. It has long been known that an optimal solution to the problem can be obtained iteratively by constructing a minimum cost network flow problem and adjusting the durations of activities corresponding to a minimum capacity cut-set. A recent paper described this method, but gave no indication of how the method could be derived. It is shown here that a linear programming formulation and its dual enables this to be done very simply.     

Probabilistic Networks and R & D Portfolio Selection

Mathematical programming methods have been suggested and used as an aid to R & D project portfolio selection. One of the main criticisms of the use of such models is that the stochastic nature of the problem has been largely ignored. This paper presents a method which takes into account the stochastic nature of resource requirements and project benefits, using a combination of probabilistic networks, simulation and mathematical programming. A case study based on data from an industrial R & D laboratory is presented and compared with the use of expected value methods. The results of the study indicate that in this particular case the deterministic linear programming solution is robust.     

Berechnung stochastischer Termine

Zusammenfassung Es wird ein neues Verfahren zur Berechnung der Wahrscheinlichkeitsverteilung der Termine bei Projekten mit zufälligen Tätigkeitsdauern hergeleitet. Indem als Näherungen für die gegebenenW-Verteilungen solche mit Elementarfunktionen alsW-Dichten genommen werden und die stochastischen Termine sequentiell berechnet werden, entsteht ein Verfahren, wie es vom deterministischen Fall her bekannt ist. Im Vergleich zu PERT ist es jedoch in allen Teilen mathematisch exakt begründbar. Andererseits ist es fast so einfach wie PERT. Insbesondere kann die Konvergenz mit feiner werdender Diskretisierung gezeigt werden. Bereits bei grober Diskretisierung liefert es jedoch praktisch brauchbare Ergebnisse, für die außerdem noch obere und untere Abschätzungen durch stochastisch kleinere bzw. größere Termine angegeben werden können. Eine Erweiterung auf die Rückwärtsrechnung ist prinzipiell möglich.

---

Summary A new method is derived for computing the probability distribution function of events in project planning with stochastic duration of the activities. Approximating the given probability distribution by elementary functions as densities and computing sequentially the stochastic events one gets a method like that in the deterministic case. In contrary to PERT it is possible to prove this method in all details with a mathematical argumentation. Moreover it is as almost easy as PERT. Especially the convergence with increasing discretization is proved. However a rough discretization already yields applicable results, moreover inferior and superior bounds by stochastically less and stochastically greater events can be computed. An extension to the backward computation is possible on principle.

     

马尔可夫骨架PERT网络的最长路径

本文利用马尔可夫骨架过程理论研究PERT网络模型，其中网络各弧线的长度是相互独立的随机变量。文中构造了一个马尔可夫骨架过程，利用其向后方程求解随机网络最长路径长度的分布函数。     

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.     

Sampling scattered data with Bernstein polynomials: stochastic and deterministic error estimates

Viewing the classical Bernstein polynomials as sampling operators, we study a generalization by allowing the sampling operation to take place at scattered sites. We utilize both stochastic and deterministic approaches. On the stochastic side, we consider the sampling sites as random variables that obey some naturally derived probabilistic distributions, and obtain Chebyshev type estimates. On the deterministic side, we incorporate the theory of uniform distribution of point sets (within the framework of Weyl’s criterion) and the discrepancy method. We establish convergence results and error estimates under practical assumptions on the distribution of the sampling sites.     

On the measurement of complexity in activity networks

The measurement of the ‘complexity’ of activity networks seems to be needed in order to estimate the computing requirements and/or to validly compare alternative heuristic procedures. This paper critically evaluates past contributions to the problem, and explores the underlying concepts of measurement. It suggests that the objective of analysis of the network is a determining factor in the process of measurement, and discusses three different objectives; they are: to determine the critical path assuming deterministic time estimates; to determine the probability distribution function of project completion assuming random durations of the activities; and to determine the optimal schedule under limited availability of a single resource. It proposes the form of the measure of network complexity for each objective, and explicitly exhibits the form of the measure relative to the first objective based on a sample of 104 networks.     

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.     

Itô’s calculus under sublinear expectations via regularity of PDEs and rough paths

In this paper, we first study the martingale problem in a sublinear expectation space. The critical tool is the Evans–Krylov theorem on regularity properties for solutions of fully nonlinear PDEs. Based on the analysis for the martingale problem and inspired by the rough path theory, we then develop stochastic calculus with respect to a general stochastic process, and derive an Itô type formula and the integration-by-parts formula. Our framework is analytic in that it does not rely on the probabilistic concept of “independence” as in the $G$-expectation theory.     

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。