Productivity Scheduling Method: Linear Schedule Analysis with Singularity Functions

This paper describes a new integrated method of linear schedule analysis using singularity functions. These functions have previously been used for structural analysis and are newly applied to scheduling. Linear schedules combine information on time and amount of work for each activity. A general model is presented with which activities and their buffers can be mathematically described in detail. The algorithm of the new method forms the body of the paper, including the steps of setting up initial equations, calculating pairwise differences between them, differentiating these to obtain the location of any minima, and deriving the final equations. The algorithm consolidates the linear schedule under consideration of all constraints and, thus, automatically generates the minimum overall project duration. The model distinguishes time and amount buffers, which bears implications for the definition and derivation of the critical path. Future research work will address float and resource analysis using the new model.     

Float Types in Linear Schedule Analysis with Singularity Functions

This paper describes how float can be calculated exactly for linear schedules by using singularity functions. These functions originate in structural engineering and are newly applied to scheduling. They capture the behavior of an activity or buffer and the range over which it applies and are extensible to an infinite number of change terms. This paper builds upon the critical path analysis for linear schedules, which takes differences between singularity functions and differentiates them. It makes several important case distinctions that extend the earlier concept of rate float. Time and location buffers act along different axis directions. Together with different productivities between and within activities, this can create a complex pattern of critical and noncritical segments. Depending on starts and finishes, areas of float precede or follow these noncritical segments. The schedule of a small project is reanalyzed with case distinctions to demonstrate in detail what float types are generated.     

Network Scheduling Variations for Repetitive Work

Observant CPM schedulers in the early years of CPM application, the 1960s, noted that there were often familiar steps inherent in scheduling similar projects. This similarity occurred in the development of housing neighborhoods, such as New Town of Columbia, Maryland; schools, as in the Philadelphia School District Program involving 200 schools; and in the review process for projects during the preconstruction phase in large cities, namely Philadelphia and New York. In these situations, the development of a prototypical network provided a cost‐effective method of applying network scheduling to major programs. These applications are described in the form of case studies, and a recent application to the King Khalid Military City Housing Program is noted. An alternate manual approach to repetitive scheduling in highrise construction, the vertical production method developed in the 1970s, is also described.     

Comparison of Linear Scheduling Model and Repetitive Scheduling Method

Various linear scheduling techniques have been proposed over the years. The limitations on these techniques have been the inability to determine critical activities. Contractors and departments of transportation have identified this need. Recently two different methods have been proposed—the linear scheduling model and the repetitive scheduling method. This paper discusses basic linear scheduling techniques and then the calculation of critical activities of basic linear scheduling elements using the two methods. The results of the two techniques are compared.     

Linear versus Network Scheduling: A Critical Path Comparison

Linear scheduling methods provide an alternative way of scheduling repetitive projects, to the commonly used network methods. Critical path identification is a major attribute for both methods; therefore, it is very important for practitioners to understand the function of the two methods in this area. The present paper compares the critical path of the recently developed Kallantzis-Lambropoulos repetitive project model against the network scheduling critical path method (CPM), aiming at delving into and pointing out the differences and similarities between them. Initially, the rules for transforming the linear project into an equivalent CPM network are proposed. Then, the rules are applied on a sample linear project. Due to the additional constraint for maintaining resource continuity that the linear method takes into account, the critical paths vary. The constraint is subsequently removed from selected activities and comparison is repeated; the critical paths then coincide. In order to validate the findings and ensure impartiality of results, a random linear project generator is developed. A group of twenty-five random linear projects and their equivalent networks is produced. Their critical paths are analyzed, compared and classified. Conclusions support that the proposed comparison could be beneficial to users of linear scheduling methods, while the random project generator can serve other related research.     

Probability of Project Completion Using Stochastic Project Scheduling Simulation

This paper introduces a software, Stochastic Project Scheduling Simulation (SPSS), developed to measure the probability to complete a project in a certain time specified by the user. To deliver a project by a completion date committed to in a contract, a number of activities need to be carried out. The time that an entire project takes to complete and the activities that determine total project duration are always questionable because of the randomness and stochastic nature of the activities’ durations. Predicting a project completion probability is valuable, particularly at the time of bidding. The SPSS finds the longest path in a network and runs the network a number of times specified by the user and calculates the stochastic probability to complete the project in the specified time. The SPSS can be used by a contractor: (1) to predict the probability to deliver the project in a given time frame and (2) to assess its capabilities to meet the contractual requirement before bidding. The SPSS can also be used by a construction owner to quantify and analyze the risks involved in the schedule. The benefits of the tool to researchers are: (1) to solve program evaluation and review technique problems; (2) to complement Monte Carlo simulation by applying the concept of project network modeling and scheduling with probabilistic and stochastic activities via a web based Java Simulation which is operateable over the Internet, and (3) to open a way to compare a project network having different distribution functions.     

Automated Statistical Analysis in Stochastic Project Scheduling Simulation

This paper describes a stochastic simulation-based scheduling system (S3) that: (1) integrates the deterministic critical path method (CPM), the probabilistic program evaluation and review technique (PERT), and the stochastic discrete event simulation (DES) approaches into a single system and lets the scheduler make an informed decision as to which method is better suited to the company’s risk-taking culture; (2) automatically determines the minimum number of simulation runs in DES mode and therefore optimizes the simulation process; and (3) provides a terminal method that tests the statistical significance of the differences between simulations, hence eliminating outliers and therefore increasing the accuracy of the DES process. The system is based on an earlier version of the system called stochastic project scheduling simulation and makes use of all the capabilities of this system. The study is of value to practitioners because S3 produces a realistic prediction of the probability of completing a project in a specified time. The study is also of relevance to researchers in that it allows researchers to compare the outcome of CPM, PERT, and DES under different conditions such as different variability or skewness in the activity duration data, the configuration of the network, or the distribution of the activity durations.     

Exact Algorithm for Solving Project Scheduling Problems under Multiple Resource Constraints

This paper presents a new algorithm, called the enumerative branch-and-cut procedure (EBAC), for minimizing the total project duration of a construction project under multiple resource constraints based on an enumeration tree. The EBAC generates new branches to the tree corresponding to “better” feasible alternatives. It starts with all of the feasible schedule alternatives as the trial schedule alternatives at any node. The trial schedule alternatives are then evaluated to determine whether they are “worse” than any existing partial schedules in the tree by using the presented cut rules, and a worse alternative will be eliminated from the enumeration tree. In other words, the tree will contain only better feasible schedules. The presented algorithm has been coded in the VB6.0 language on a personal computer. It has been tested with the 110 scheduling problems, which have been widely used for validating a variety of schedule algorithms over the last 20?years. The EBAC can obtain the shortest project durations for all of the 110 problems.     

Transform Schemes Applied on Non-Finish-to-Start Logical Relationships in Project Network Diagrams

Critical path analysis on a project network having non-finish-to-start (FS) logical relationships with lags is generally referred to as precedence diagram method (PDM). A PDM-based scheduling analysis is facilitated by mainstream project scheduling software (such as P3). However, PDM compounds total float determination and interpretation, potentially causing anomalous effects on critical path identification. In the present research, we generalize those particular circumstances that entail applying non-FS logical relationships on construction projects. We then propose generic transform schemes such that non-FS relationships in a PDM network can be detected and transformed—automatically—into equivalent FS. Moreover, we provide analytical proofs for the transform schemes being proposed to justify the logical equivalency between the original PDM network and the transformed activity-on-node (AON) network only having FS logical relationships. A PDM network example demonstrates that confusions would arise in interpreting P3’s critical path analysis results, but not in the case of the transformed AON counterpart. In conclusion, the transform schemes being proposed lead to better understanding of the scheduling results when critical path analysis is performed on a PDM network. This also paves the way for conducting further sophisticated scheduling analysis (such as resource loading or Monte Carlo simulation) on a PDM network.     

Stochastic Network Model for Planning Scheduling

A stochastic network model consisting of dependent and independent random variables is developed for construction scheduling. The network model is based on Monte‐Carlo simulation. Data for each network activity consist of a time distribution for the activity under optimal conditions and a series of time distributions for various problems that may lengthen the activity completion time. Dependencies between network activities may be modelled; also, time dependencies for a network activity may be modelled. The implementation of the model is discussed.     

Critical Path Segments Scheduling Technique

While the critical path method (CPM) has been useful for scheduling construction projects, years of practice and research have highlighted serious drawbacks that hinder its use as a decision support tool. This paper argues that many of CPM drawbacks stem from the rough level of detail at which the analysis is conducted, where activities’ durations are considered as continuous blocks of time. The paper thus proposes a new critical path segments (CPS) mechanism with a finer level of granularity by decomposing the duration of each activity into separate time segments. Three cases are used to prove the benefits of using separate time segments in avoiding complex network relationships, accurately identifying all critical path fluctuation, better allocation of limited resources, avoiding multiple-calendar problems, and accurate analysis of project delays. The paper discusses the proposed CPS mechanism and comments on several issues related to its calculation complexity, its impact on existing procedures, and future extensions. This research is more beneficial to researchers and has the potential to revolutionize scheduling computations to resolve CPM drawbacks.     

Work Continuity Constraints in Project Scheduling

Repetitive projects involve the repetition of activities along the stages of the project. Since the resources required to perform these activities move from one stage to the other, a main objective of scheduling these projects is to maintain the continuity of work of these resources so as to minimize the idle time of resources. This requirement, often referred to as work continuity constraints, involves a tradeoff between total project duration and the resource idle time. The contribution of this paper is threefold. First, we provide an extensive literature summary of the topic under study. Although most research papers deal with the scheduling of construction projects, we show that this can be extended to many other environments. Second, we propose an exact search procedure for scheduling repetitive projects with work continuity constraints. This algorithm iteratively shifts repeating activities further in time in order to decrease the resource idle time. We have embedded this recursive search procedure in a horizon-varying algorithm in order to detect the complete tradeoff profile between resource idle time and project duration. The procedure has been coded in Visual C++ and has been validated on a randomly generated problem set. Finally, we illustrate the concepts on three examples. First, the use of our new algorithm is illustrated on a small fictive problem example from literature. In a second example, we show that work continuity constraints involve a tradeoff between total project duration and the resource idle time. A last example describes the scheduling of a well-known real-life project that aims at the construction of a tunnel at the Westerschelde in The Netherlands.     

Evaluation of the Resource-Constrained Critical Path Method Algorithms

This study evaluates the resource-constrained critical path method (RCPM), which the writers have recently proposed. RCPM establishes a critical path method (CPM)-like, resource-constrained schedule by resource-dependent activity relationships (or resource links) that the five-step RCPM technique identifies. With its CPM-like feature, RCPM provides the critical path and float data that are not available in traditional resource-constrained scheduling techniques. In addition, RCPM provides more flexibility to the schedule through identified alternative schedules, which allow certain activities to be executed beyond their late finish times without delaying the project completion. This paper evaluates the RCPM’s performance by comparing it with five related previous studies. A brief review of each study is also included in this paper. This comparison shows that RCPM performs well in identifying resource links and alternative schedules, compared to other methods. This study is of interest to academics because it highlights the advantages and disadvantages of different algorithms that have attempted to overcome present problems in traditional resource-constrained scheduling techniques.     

New Method for Calculating Path Float in Program Evaluation and Review Technique (PERT)

The path float use in the network is an effective method used to cope with various uncertainties existing within the construction. However, the current path float calculating method may bring misleading information to the managers on site and then cause project duration risk in the construction. The purpose of this paper is to present a new method, which calculates the noncritical path float in the program evaluation and review technique (PERT), to copy with the uncertainties within the network implementation, and to reduce the misleading information. An example network was analyzed with the new method, the results showed the consistent path float under required completion probability and required duration. The new path float concept will bring useful planning information to the managers and the planners in the construction.     

Resource-Activity Critical-Path Method for Construction Planning

In this paper, a practical method is developed in an attempt to address the fundamental matters and limitations of existing methods for critical-path method (CPM) based resource scheduling, which are identified by reviewing the prior research in resource-constrained CPM scheduling and repetitive scheduling. The proposed method is called the resource-activity critical-path method (RACPM), in which (1) the dimension of resource in addition to activity and time is highlighted in project scheduling to seamlessly synchronize activity planning and resource planning; (2) the start/finish times and the floats are defined as resource-activity attributes based on the resource-technology combined precedence relationships; and (3) the “resource critical” issue that has long baffled the construction industry is clarified. The RACPM is applied to an example problem taken from the literature for illustrating the algorithm and comparing it with the existing method. A sample application of the proposed RACPM for planning a footbridge construction project is also given to demonstrate that practitioners can readily interpret and utilize a RACPM schedule by relating the RACPM to the classic CPM. The RACPM provides schedulers with a convenient vehicle for seamlessly integrating the technology/process perspective with the resource use perspective in construction planning. The effect on the project duration and activity floats of varied resource availability can be studied through running RACPM on different scenarios of resources. This potentially leads to an integrated scheduling and cost estimating process that will produce realistic schedules, estimates, and control budgets for construction.     

Criticism of CPM for Project Planning Analysis

Published criticism in recent years concerning the inadequacy of Critical Path Method (CPM) as a project planning tool is identified and grouped under six major headings with reference to the publications in which the criticism were contained. These are answered from the writer's field experience and from experiences published by other authors. The object of the analysis is to see whether or not CPM as a project planning tool can meet the required functions of planning in construction, including consideration of legal and contractual framework and the complex and interdisciplinary nature of the project environment. The analysis reveals that, despite numerous criticism, project and construction planning should be done using CPM scheduling. Main factors affecting successful planning are realistic estimation of the productivity of crews in the context of expected job‐management efficiency conditions, and inclusion of sufficient time buffers between dissimilar trades. CPM is found to be equally useful as a planning tool for linear or repetitive projects. The limitations of this technique are identified in terms of the defined planning functions in the engineering phase of capital projects. A broad model for management of the engineering phase in revenue‐generating projects is suggested.     

Integrating Efficient Resource Optimization and Linear Schedule Analysis with Singularity Functions

Resources perform or enable physical operations and thus are vital on construction projects, yet are subject to various constraints. Their use within a project schedule must therefore be carefully planned. A major objective is optimizing when they are active within the float of noncritical activities to avoid disruptive and costly fluctuations. This paper builds on analyzing criticality of linear schedules with the unique singularity functions. The new approach keeps resources intact and derives one flexible equation for the complete resource profile of a schedule, including any timing or resource rate changes. Another equation models its first moment of area to minimize the objective function toward a level profile. A genetic algorithm is suitable for an iterative optimization. The parameters of its chromosomes are recombined evolutionarily and can model any permutation. Analyzing a road project illustrates how singularity functions integrate resource optimization with its linear schedule and facilitate a subsequent optimization.     

Construction Project Scheduling with Time, Cost, and Material Restrictions Using Fuzzy Mathematical Models and Critical Path Method

This article evaluates the viability of using fuzzy mathematical models for determining construction schedules and for evaluating the contingencies created by schedule compression and delays due to unforeseen material shortages. Networks were analyzed using three methods: manual critical path method scheduling calculations, Primavera Project Management software (P5), and mathematical models using the Optimization Programming Language software. Fuzzy mathematical models that allow the multiobjective optimization of project schedules considering constraints such as time, cost, and unexpected materials shortages were used to verify commonly used methodologies for finding the minimum completion time for projects. The research also used a heuristic procedure for material allocation and sensitivity analysis to test five cases of material shortage, which increase the cost of construction and delay the completion time of projects. From the results obtained during the research investigation, it was determined that it is not just whether there is a shortage of a material but rather the way materials are allocated to different activities that affect project durations. It is important to give higher priority to activities that have minimum float values, instead of merely allocating materials to activities that are immediately ready to start.     

Models for Accurate Computation of Earliest and Latest Start Times and Optimal Compression in Project Networks

Operations research techniques, especially linear and integer programming, have been suggested for use in project management. Most frequently these techniques are applied to finding critical path(s) in project networks and for compressing activities for an early project completion. The simplest of these models are usually found in introductory operations research/management science textbooks and they often suffer from a common deficiency: when they employ earliest start times (ES) as variables in the model, ES of activities that are not on the critical path may be computed incorrectly. This may lead to inefficient resource reallocation and leveling. Furthermore, activities outside the critical path may be compressed unnecessarily, as will be demonstrated in this paper. We present a zero–one mixed integer programming model to correctly compute earliest and latest start times of all activities and determine the optimal compression schedule. Several extensions to the model are offered to provide different perspectives of the project and enable a project manager to answer “what if” type questions. A small example is used to demonstrate the effectiveness of the models.     

Fast and Accurate Risk Evaluation for Scheduling Large-Scale Construction Projects

This paper presents the development of a novel probabilistic scheduling model that enables fast and accurate risk evaluation for large-scale construction projects. The model is designed to overcome the limitations of existing probabilistic scheduling methods, including the inaccuracy of the program evaluation and review technique (PERT) and the long computational time of the Monte Carlo simulation method. The model consists of three main modules: PERT model; fast and accurate multivariate normal integral method; and a newly developed approximation method. The new approximation method is designed to focus the risk analysis on the most significant paths in the project network by identifying and removing insignificant paths that are either highly correlated or have high probability of completion time. The performance of the new model is analyzed using an application example. The results of this analysis illustrate that the new model was able to reduce the computational time for a large-scale construction project by more than 94% while keeping the error of its probability estimates to less than 3%, compared with Monte Carlo Simulation methods.