灰色网络计划中的关键线路确定方法研究

由于目前区间灰数的表征和运算过程存在着运算结果对灰数的灰度进行不必要的放大,造成信息的严重失真的问题,所以在灰色网络计划中计算出的工作总时差也有不同程度的放大,这时再根据工作总时差为零或最小的方法来确定关键线路则是不准确的.给出了确定灰色网络计划关键线路可以根据关键线路上总工期最长的方法,简单易行,且有效避免了灰数运算对灰数灰度放大的问题.     

基于S型隶属度函数的模糊网络计划关键线路分析

在众多研究三角或梯形模糊网络计划的基础上,分析了S型隶属函数下模糊网络计划的可行性.针对在模糊网络计划中以总时差为零来判断关键线路可能遇到的问题,引出相关面积的概念,以最接近总工期的线路对工程工期影响最大的原则,得出关键线路的求法.     

求解CPM网络计划的最大网络时差

CPM网络计划的网络时差表示项目中各工序实际可使用的机动时间的总和(绝非理论上机动时间的简单加总),即CPM网络计划的总机动时间,它决定着在总工期不变的前提下,所有工序实际可以达到的最大工期的总和,与项目的成本管理和时间管理密切相关。网络时差是变量,取决于各工序的时间进度安排,说明可以通过调整工序的时间进度来决定该时差的取值,特别是其最大值,进而实现成本和时间优化。本文首先从新的角度分析了网络时差的含义;然后,在此基础上设计了求解最大网络时差的算法,其思路为,通过建立和分析最大网络时差模型,将其转化为特殊的“时间-费用权衡问题”,进而可运用Fulkerson算法等经典算法求解;最后,通过应用举例对该算法进行了演示。     

Criticality in Resource Constrained Networks

Project managers readily adopted the concept of the critical path as an aid to identifying those activities most worthy of their attention and possible action. However, current project management packages do not offer a useful measure of criticality in resource constrained projects. A revised method of calculating resource constrained float is presented, together with a discussion of its use in project management. While resource constrained criticality appears to be a practical and useful tool in the analysis of project networks, care is needed in its interpretation as any calculation of such float is conditional on the particular resource allocation employed. A number of other measures of an activity's importance in a network are described and compared in an application to an aircraft development. A quantitative comparison of the measures is developed based on a simulation of the process of management identifying the key activities and directing their control efforts. Resource constrained float appears to be a useful single measure of an activity's importance, encapsulating several useful pieces of management information. However, there are some circumstances in which other measures might be preferred.     

等效化简带有广义优先关系的时间-费用权衡问题

对于经典的时间-费用权衡问题,工序之间只存在单一时间约束,可用CPM网络表示。但是对于工序之间存在多种时间约束的时间-费用权衡问题,包括最大和最小时间约束(称为广义优先关系,简称GPRs),则只能用GPRs网络表示,比CPM网络复杂许多。首先,论述了带有GPRs的时间-费用权衡问题与经典问题的巨大差别:在GPRs中,(1)缩短某些关键工序的工期能使总工期缩短,但缩短另一些关键工序的工期反而能使总工期延长;(2)缩短或延长工序的工期可能会破坏项目自身的可行性;等。其次,研究了GPRs网络的特性,推导出该网络的路长定理。第三,根据该定理,设计出等效化简带有GPRs的大型时间-费用权衡问题的简单方法,从而大幅减小求解该问题的难度和计算量。最后,通过算例演示了该方法。     

基于未确知有理数的施工网络研究

提出未确知有理数用于施工网络计划,介绍未确知有理数概念、运算、未确知期望.建立未确知施工网络计划,提出未确知施工网络计划关键线路计算模型,确定关键线路和未确知施工工期.讨论不确定性网络计划的研究.     

Multiple schedules and measures of resource constrained float

The conventional analysis of a resource constrained network produces a single schedule. A measure of float for each activity can be defined by examining the sequences of activities, incorporating the linkages implied by the sharing of resources in this particular schedule. However, the resultant floats are specific to one schedule and the analysis ignores the flexibility inherent in many resource constrained networks. It is often possible to employ alternative resource allocations resulting in schedules with identical overall durations but different timings for individual activities; in one schedule an activity may be critical but in another it may have significant float. A systematic exploration of alternative schedules reveals the flexibility of the network and the consequent variation in activities' floats. This paper defines new parameters describing these characteristics of activities in resource constrained networks. The practical value of the parameters is demonstrated in an example in which the measures of float guide the project planner to an alternative schedule, avoiding a critical dependency on a problematic activity and reducing the risk to the project at no additional cost.     

A simple approach to fuzzy critical path analysis in project networks

This paper develops a simple approach to critical path analysis in a project network with activity times being fuzzy numbers. The idea is based on the linear programming (LP) formulation and fuzzy number ranking method. The fuzzy critical path problem is formulated as an LP model with fuzzy coefficients of the objective function, and then on the basis of properties of linearity and additivity, the Yager’s ranking method is adopted to transform the fuzzy LP formulation to the crisp one which can be solved by using the conventional streamlined solution methods. Consequently, the critical path and total duration time can be obtained from the derived optimal solution. Moreover, in this paper we also define the most critical path and the relative path degree of criticality, which are theoretically sound and easy to use in practice. An example discussed in some previous studies illustrates that the proposed approach is able to find the most critical path, which is proved to be the same as that derived from an exhausted comparison of all possible paths. The proposed approach is very simple to apply, and it is not require knowing the explicit form of the membership functions of the fuzzy activity times.     

Cost impact of float loss on a project with adjustable activity durations

Although delays to non-critical activities within the float do not always affect the overall completion time of a project, they commonly cause disputes over the impact cost and apportionment resulting from the complexity of resource utilization in construction projects. Therefore, considerable attention has been focused on providing an effective and reliable method for analysing the effects of float loss. Several recent studies have proposed various methods; however, most of these methods are based on the assumption of a fixed duration for each activity or activity-based cost simulation. Few studies have considered the trade-off between time and costs and the integration of project resources. Using genetic algorithms, this study introduces a critical path method (CPM)-modified resource-integrated optimization model and successfully quantifies the impact of float loss on the total cost of the project. The results provide objective quantification for accurately evaluating the impact of within-float delays and facilitate the analysis of the impact of delay claims on cost and apportionment in construction projects.     

Persistence in discrete optimization under data uncertainty

An important question in discrete optimization under uncertainty is to understand the persistency of a decision variable, i.e., the probability that it is part of an optimal solution. For instance, in project management, when the task activity times are random, the challenge is to determine a set of critical activities that will potentially lie on the longest path. In the spanning tree and shortest path network problems, when the arc lengths are random, the challenge is to pre-process the network and determine a smaller set of arcs that will most probably be a part of the optimal solution under different realizations of the arc lengths. Building on a characterization of moment cones for single variate problems, and its associated semidefinite constraint representation, we develop a limited marginal moment model to compute the persistency of a decision variable. Under this model, we show that finding the persistency is tractable for zero-one optimization problems with a polynomial sized representation of the convex hull of the feasible region. Through extensive experiments, we show that the persistency computed under the limited marginal moment model is often close to the simulated persistency value under various distributions that satisfy the prescribed marginal moments and are generated independently.     

基于时差分析的时标网络图探究

鉴于已有时差概念并不能充分反映CPM网络紧前和紧后工序时差的内在联系，文中引入三个新时差（前共后单时差，前单后共时差，双共时差）概念。针对目前工程项目管理中通用的时标网络图一早时标网络图和迟时标网络图，在路长定理的基础上，给出其理论依据，并在时差分析的基础上，随机绘出时标网络图，三个新时差的引入将有助于今后时标网络图的绘制，使网络技术的应用更具灵活性和实用性。     

Discrete time/cost trade-offs in project scheduling with time-switch constraints

The discrete time/cost trade-off problem assumes the duration of project activities to be discrete, non-increasing functions of the amount of a single non-renewable resource. The problem has been studied under three possible objectives. The so-called deadline problem involves the scheduling of project activities in order to minimize the total cost of the project while meeting a given deadline. The budget problem aims at minimizing the project duration without exceeding a given budget. A third objective involves the generation of the complete efficient time/cost profile over the set of feasible project durations. In this paper we describe a solution procedure for the deadline problem in which three special cases of time-switch constraints are involved. These constraints impose a specified starting time on the project activities and force them to be inactive during specified time periods. We propose a branch-and-bound algorithm and a heuristic procedure which both make use of a lower bound calculation for the discrete time/cost trade-off problem (without time-switch constraints). The procedures have been coded in Visual C++, version 6.0 under Windows 2000 and have been validated on a randomly generated problem set. We also discuss an illustrative example based on a real-life situation.     

Markov skeleton process in PERT networks

In this article, we investigate Programming Evaluation and Review Technique networks with independently and generally distributed activity durations. For any path in this network, we select all the activities related to this path such that the completion time of the sub-network (only consisting of all the related activities) is equal to the completion time of this path. We use the elapsed time as the supplementary variables and model this sub-network as a Markov skeleton process, the state space is related to the sub-network structure. Then use the backward equation to compute the distribution of the sub-network's completion time, which is an important rule in project management and scheduling.     

MARKOV SKELETON PROCESS IN PERT NETWORKS

In this article, we investigate Programming Evaluation and Review Technique networks with independently and generally distributed activity durations. For any path in this network, we select all the activities related to this path such that the completion time of the sub-network （only consisting of all the related activities） is equal to the completion time of this path. We use the elapsed time as the supplementary variables and model this sub-network as a Markov skeleton process, the state space is related to the subnetwork structure. Then use the backward equation to compute the distribution of the sub-network＇s completion time, which is an important rule in project management and scheduling.     

重复性建设项目中确定关键路线的方法研究

本文提出了一种确定重复性建设项目关键路线的新方法。借助约束线,首先给出了工序间存在各种约束条件(时间和距离约束)下潜在关键点的确定方法;为处理大规模项目,进一步提出了与图示法相对应的数值算法。以此为基础,提出了确定关键工序和关键路线的具体步骤,并定义和分析了三种不同类型的关键工序。与现有的方法相比,本文提出的确定关键路线的方法更为准确,适用性更强,而且有利于调度优化目标的实现。     

A new approach to find project characteristics and multiple possible critical paths in a fuzzy project network

Graphs are widely used to represent data and relationships. Among all graphs, a particularly useful family is the family of trees. In this paper, we utilize a rooted tree to describe a fuzzy project network as it enables simplification in finding earliest starting times and trapezoidal fuzzy numbers to express the operation times for all activities in project network. As there is an increasing demand that the decision maker needs “Multiple possible critical paths” to decrease the decision risk for project management, in this paper, we introduce an effective graphical method to compute project characteristics such as total float, earliest and latest times of activities in fuzzy project network and a new ranking to find possible critical paths. Numerical example is provided to explain the proposed procedure in detail; the results have shown that the procedure is very useful and flexible in finding total floats. By comparing the critical paths obtained by this method with the previous methods, it is shown that the proposed method is effective in finding possible critical paths.     

用最少的虚工序构建等效多阶段工序网络

运用网络计划可以直观地表示项目管理中的诸多疑难问题, 便于分析和求解. 但是它也存在明显的缺点, 如, (1) 工序网络的有向无回路性表明很多时候适合运用动态规划法, 但它在通常情况下的无阶段性使得该方法无法直接应用; (2) 任意构建的工序网络容易表现得错综复杂, 不利于研究; (3) 用最少的虚工序表示双代号网络是NP-难问题, 因此对一个工序系统可能构建出多个差别迥异的工序网络, 有碍于进度计划管理研究, 等等. 如果能将工序网络构建成等效的多阶段网络, 各工序分别表示在相应的阶段中, 无疑有助于上述问题的解决. 构建等效多阶段工序网络需要添加虚工序. 通过添加最少的虚工序将工序网络构建成等效多阶段网络, 从而有助于建立更合理的工序网络表示法.     

Analysis of critical paths in a project network with fuzzy activity times

This paper proposes an approach to critical path analysis for a project network with activity times being fuzzy numbers, in that the membership function of the fuzzy total duration time is constructed. The basic idea is based on the extension principle and linear programming formulation. A pair of linear programs parameterized by possibility level α is formulated to calculate the lower and upper bounds of the fuzzy total duration time at α. By enumerating different values of α, the membership function of the fuzzy total duration time is constructed, and the fuzzy critical paths are identified at the same time. Moreover, by applying the Yager ranking method, definitions of the most critical path and the relative degree of criticality of paths are developed; and these definitions are theoretically sound and easy to use in practice. Two examples with activity times being fuzzy numbers of L-R and L-L types discussed in previous studies are solved successfully to demonstrate the validity of the proposed approach. Since the total duration time is completely expressed by a membership function rather than by a crisp value, the fuzziness of activity times is conserved completely, and more information is provided for critical path analysis.     

A note on activity floats in activity-on-arrow networks

In project scheduling, the free float (slack) of an activity represents the leeway for scheduling the activity without affecting any subsequent activity. In the context of activity-on-arrow (AOA) network representation, textbooks on project management, based on our survey, have for decades been using a popular formula to calculate free floats that may lead to erroneous results in the presence of dummy arcs. In this note, we present a correct version of the formula. Extensions of the proposed formula to other floats such as safety float and interference float are also discussed.     

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.