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Impact of Uncertainty on Construction Project Performance Using Linear Scheduling

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Title:
Impact of Uncertainty on Construction Project Performance Using Linear Scheduling
Creator:
TROFIN, IULIAN ( Author, Primary )
Copyright Date:
2008

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Subjects / Keywords:
Arithmetic mean ( jstor )
Construction engineering ( jstor )
Construction industries ( jstor )
Line of balance technique ( jstor )
Linear programming ( jstor )
Linear scheduling ( jstor )
Research methods ( jstor )
Scheduling ( jstor )
Simulations ( jstor )
Standard deviation ( jstor )

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University of Florida
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University of Florida
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Copyright Iulian Trofin. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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4/30/2005
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436097539 ( OCLC )

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IMPACT OF UNCERTAINTY ON CONSTRUCTION PROJECT PERFORMANCE USING LINEAR SCHEDULING By IULIAN TROFIN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN BUILDING CONSTRUCTION UNIVERSITY OF FLORIDA 2004

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Copyright 2004 by Iulian Trofin

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ACKNOWLEDGMENTS First and foremost, I would like to acknowledge my committee members, Dr. Ian Flood, Dr. Raymond Issa, and Dr Robert Cox, for their time and assistance through the completion of this thesis. A special acknowledgement goes to my committee chair, Dr. Ian Flood, for his practical guidance and unwavering encouragement. He helped me to further my educational advancement by setting high standards of excellence and challenging me to learn and develop in the building construction field. In addition, I would like to thank my friends. Their moral support and practical help have strengthened my life in Gainesville. My deepest gratitude goes to my family and especially to my mother, for selflessly providing me continuous support and unconditional love. I would never be where I am without the sacrifices they have made for me. Their confidence in me motivates and supports me in pursuing my goals in every way. iii

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TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iii LIST OF TABLES.............................................................................................................vi LIST OF FIGURES...........................................................................................................ix ABSTRACT......................................................................................................................xii CHAPTER 1 INTRODUCTION........................................................................................................1 2 LITERATURE REVIEW.............................................................................................3 Linear Scheduling Method (LSM)................................................................................3 Research Efforts for Linear Scheduling Method...................................................3 Other Approaches to Linear Scheduling...............................................................7 Comparison of LSM and CPM.....................................................................................9 Examples LSM/CPM..................................................................................................10 Line of Balance (LOB)...............................................................................................13 Disadvantages of Line of Balance (LOB) and Critical Path Method (CPM).................................................................................................17 Attempts of Combining CPM and LOB..............................................................18 Stochastic/Probabilistic Approach in Project Scheduling..........................................19 3 RESEARCH OBJECTIVES AND METHODOLOGY.............................................23 Objectives...................................................................................................................23 Methodology: Generating Data..................................................................................24 Deterministic Project Duration............................................................................25 Activity Variance.................................................................................................31 Activity Idle Time...............................................................................................33 Project Idle Time.................................................................................................36 Methodology: Data Analysis......................................................................................43 4 ANALYSIS OF RESULTS........................................................................................44 Project Duration and Standard Deviation...................................................................44 iv

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Project Duration...................................................................................................44 Standard Deviation..............................................................................................46 Tendencies in the Project Duration Values................................................................47 Probability of Occurrence...........................................................................................49 Activity Idle Time and Project Idle Time...................................................................50 Variation of Activity Idle and Project Idle..........................................................51 Activity Idle Time Savings..................................................................................53 Amount of Delay in Project Duration.................................................................54 5 CONCLUSIONS AND RECOMMENDATIONS.....................................................56 Conclusions.................................................................................................................56 Recommendations for Future Research......................................................................58 APPENDIX A DETERMINISTIC PARAMETERS FOR PROJECT #1 THROUGHOUT PROJECT #30............................................................................................................60 B DETERMINISTIC LINEAR SCHEDULES..............................................................71 LIST OF REFERENCES...................................................................................................87 BIOGRAPHICAL SKETCH.............................................................................................91 v

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LIST OF TABLES Table page 2-1 Linear scheduling—Methods in the construction industry....................................11 3-1 Activity production rates........................................................................................25 3-2 Production rate values for Project #1 to Project #5...............................................27 3-3 Production rate values for Project #6 to Project #10.............................................27 3-4 Production rate values for Project #11 to Project #15...........................................28 3-5 Production rate values for Project #16 to Project #20...........................................28 3-6 Production rate values for Project #21 to Project #25...........................................28 3-7 Production rate values for Project #26 to Project #30...........................................29 3-8 Production rates, activity duration, start time and finish time for Project #1...............................................................................................................29 3-9 Linear schedule values...........................................................................................32 3-10 Linear schedule values (deterministic start time and not delayed finish time).............................................................................................................34 3-11 Linear schedule values (rescheduled)....................................................................35 3-12 Linear schedule values (deterministic)..................................................................36 3-13 Linear schedule values (not rescheduled)..............................................................37 3-14 Linear schedule values (rescheduled)....................................................................38 3-15 Mean values of the trials for RiskNormal (0, 10%) function................................40 3-16 Mean values of the trials for RiskNormal (0, 20%)...............................................41 3-17 Mean values of the trials for RiskNormal (0, 30%)...............................................42 4-1 Averages of project duration (days).......................................................................45 vi

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4-2 Averages of standard deviation (days)...................................................................46 4-3 Project duration and standard deviation values.....................................................48 4-4 Variation of activity idle and project duration.......................................................52 4-5 Saves in activity idle..............................................................................................54 4-6 Amount of delay in project duration......................................................................55 A-1 Production rates, activities duration, start and finish time for Project #1..............60 A-2 Production rates, activities duration, start and finish time for Project #2..............60 A-3 Production rates, activities duration, start and finish time for Project #3..............61 A-4 Production rates, activities duration, start and finish time for Project #4..............61 A-5 Production rates, activities duration, start and finish time for Project #5..............61 A-6 Production rates, activities duration, start and finish time for Project #6..............62 A-7 Production rates, activities duration, start and finish time for Project #7..............62 A-8 Production rates, activities duration, start and finish time for Project #8..............62 A-9 Production rates, activities duration, start and finish time for Project #9..............63 A-10 Production rates, activities duration, start and finish time for Project #10............63 A-11 Production rates, activities duration, start and finish time for Project #11............63 A-12 Production rates, activities duration, start and finish time for Project #12............64 A-13 Production rates, activities duration, start and finish time for Project #13............64 A-14 Production rates, activities duration, start and finish time for Project #14............64 A-15 Production rates, activities duration, start and finish time for Project #15............65 A-16 Production rates, activities duration, start and finish time for Project #16............65 A-17 Production rates, activities duration, start and finish time for Project #17............65 A-18 Production rates, activities duration, start and finish time for Project #18............66 A-19 Production rates, activities duration, start and finish time for Project #19............66 A-20 Production rates, activities duration, start and finish time for Project #20............66 vii

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A-21 Production rates, activities duration, start and finish time for Project #21............67 A-22 Production rates, activities duration, start and finish time for Project #22............67 A-23 Production rates, activities duration, start and finish time for Project #23............67 A-24 Production rates, activities duration, start and finish time for Project #24............68 A-25 Production rates, activities duration, start and finish time for Project #25............68 A-26 Production rates, activities duration, start and finish time for Project #26............68 A-27 Production rates, activities duration, start and finish time for Project #27............69 A-28 Production rates, activities duration, start and finish time for Project #28............69 A-29 Production rates, activities duration, start and finish time for Project #29............69 A-30 Production rates, activities duration, start and finish time for Project #30............70 viii

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LIST OF FIGURES Figure page 2-1 Linear schedule (Source: Peer, 1974)......................................................................5 2-2 Optimum linear schedule for a bridge construction project (Source: Selinger, 1980)........................................................................................................................8 3-1 LogLogistic distribution.........................................................................................26 3-2 Linear schedule for Project #1(deterministic values)............................................30 3-3 Linear schedule (actual production rates)..............................................................33 3-4 Linear schedule (not delayed finish time)..............................................................34 3-5 Linear schedule (rescheduled)...............................................................................35 3-6 Linear schedule for Project #2 (deterministic values)...........................................36 3-7 Linear schedule (not-rescheduled).........................................................................38 3-8 Linear schedule (rescheduled)...............................................................................39 4-1 Average project duration........................................................................................46 4-2 Standard deviation average....................................................................................47 4-3 Probability of occurrence for Project #10 (10% uncertainty)................................50 4-4 Probability of occurrence for Project #10 (20% uncertainty)................................50 4-5 Probability of occurrence for Project #10 (30% uncertainty)................................51 4-6 Activity idle and project idle variation for Project #1(10% uncertainty)..............52 4-7 Activity idle and project idle variation for Project #1 (20% uncertainty).............53 4-8 Activity idle and project idle variation for Project #1 (30% uncertainty).............53 4-9 Saves in activity idle..............................................................................................54 ix

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4-10 Delays in project duration......................................................................................55 B-1 Deterministic linear schedule Project #1...............................................................71 B-2 Deterministic linear schedule Project #2...............................................................72 B-3 Deterministic linear schedule Project #3...............................................................72 B-4 Deterministic linear schedule Project #4...............................................................73 B-5 Deterministic linear schedule Project #5...............................................................73 B-6 Deterministic linear schedule Project #6...............................................................74 B-7 Deterministic linear schedule Project #7...............................................................74 B-8 Deterministic linear schedule Project #8...............................................................75 B-9 Deterministic linear schedule Project #9...............................................................75 B-10 Deterministic linear schedule Project #10.............................................................76 B-11 Deterministic linear schedule Project #11.............................................................76 B-12 Deterministic linear schedule Project #12.............................................................77 B-13 Deterministic linear schedule Project #13.............................................................77 B-14 Deterministic linear schedule Project #14.............................................................78 B-15 Deterministic linear schedule Project #15.............................................................78 B-16 Deterministic linear schedule Project #16.............................................................79 B-17 Deterministic linear schedule Project #17.............................................................79 B-18 Deterministic linear schedule Project #18.............................................................80 B-19 Deterministic linear schedule Project #19.............................................................80 B-20 Deterministic linear schedule Project #20.............................................................81 B-21 Deterministic linear schedule Project #21.............................................................81 B-22 Deterministic linear schedule Project #22.............................................................82 B-23 Deterministic linear schedule Project #23.............................................................82 B-24 Deterministic linear schedule Project #24.............................................................83 x

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B-25 Deterministic linear schedule Project #25.............................................................83 B-26 Deterministic linear schedule Project #26.............................................................84 B-27 Deterministic linear schedule Project #27.............................................................84 B-28 Deterministic linear schedule Project #28.............................................................85 B-29 Deterministic linear schedule Project #29.............................................................85 B-30 Deterministic linear schedule Project #30.............................................................86 xi

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Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science in Building Construction IMPACT OF UNCERTAINTY ON CONSTRUCTION PROJECT PERFORMANCE USING LINEAR SCHEDULING By Iulian Trofin May 2004 Chair: Ian Flood Cochair: R. Raymond Issa Major Department:Building Construction The Linear Scheduling Method is a technique that offers a practical way to model linear projects as well as an efficient framework to monitor their progress. It not only allows the positioning of activities in a time/space format but also presents a method to determine the controlling activity path for the project. The production rates for the activities are also included in the schedule in the form of the slope of the lines that represent the activities. Unfortunately, all the research conducted until now has been undertaken using a deterministic approach to construction project scheduling. Uncertainty in the duration of an activity is commonplace in the construction industry. This study was undertaken in order to assess how the performance of a construction project is affected by different degrees of uncertainty, and to determine whether the optimal schedule for a project will differ from that produced using a deterministic analysis. xii

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In order to generate data used in this study, several simulations were conducted using data provided by a construction project situated in northern Michigan, namely the widening of a segment of U.S. Route 41. The Palisade BestFit computer program was used in order to fit the available data to statistical distributions, and the Palisade MonteCarlo@Risk software was used to introduce variances (uncertainty) to the activities of the linear schedule. A deterministic set of results (when uncertainty was not considered) and a stochastic set of results (when variance was introduced to the activities duration) were thus obtained. The results of the deterministic approach were analyzed and evaluated against the results of the stochastic approach in order to determine the impact of uncertainty on construction project performance. At the same time, an analysis of how the level of uncertainty affects the project parameters (project duration, project idle time, activity idle time) was conducted. This was done using detailed graphs of the parameters involved. Also, a solution to determine the optimum solution in terms of project parameters was presented. This study concluded that in the construction industry, uncertainty is a very important factor and has to be considered when a project is scheduled. When the extent of uncertainty is increased, the expected duration of the project is also increased. In addition, the deterministic project duration is unlikely to be the optimum schedule in terms of meeting the project time objectives. xiii

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CHAPTER 1 INTRODUCTION In the construction industry, repetitive projects that contain identical or similar units, such as floors in multistory buildings, houses in housing developments, linear feet in pipelines, or stations in highways, are encountered very often. For these types of projects, using the Critical Path Method or the Bar Chart Method in scheduling the activities is not very suitable. The Linear Scheduling Method was developed in order to better represent these types of construction projects. As a very simple definition, the Linear Scheduling Method represents in diagrammatic form the location and time at which a certain crew will be working throughout the project. The Linear Scheduling Method is one of the scheduling techniques that offer a practical way to model linear projects as well as an efficient framework to monitor their progress. It allows for positioning the activities in a time and space format, and the production rates for the activities are also included in the schedule in the form of the slope of the lines that represent the activities. The Linear Scheduling Method has been used in construction projects for many years, but largely using the deterministic approach which provides the planner with only the expected output of the project duration, with no representation of uncertainty in performance. Lately, researchers have concentrated their efforts in approaching the construction scheduling in a stochastic view, using uncertainty in considering the activities 1

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2 performance. However, this approach was not used in the case of linear construction projects, and therefore this study will serve to show its significance. The activity production rate should reflect the real behavior of the activity in order to present the construction operation in a realistic form. This goal has been achieved by introducing variances for the activities duration. This research study was conducted in order to demonstrate that in the case of linear scheduling for construction, a deterministic approach is not necessarily the optimal solution in terms of project performances. By adopting a stochastic approach and considering the uncertainties in a construction project, the results obtained will represent a more realistic solution in terms of project duration. In the same time, this study will show the impact of uncertainty on important project performance parameters as project idle time and activity idle time. This introduction of the study is the first of five chapters to address the different parts of this research. The literature review in Chapter 2 presents in detail all the available methods used in linear scheduling in the construction industry and previous research completed in this area. Chapter 3 describes research objectives and methods and defines the goals of this research, and also discusses the procedures and software used to complete this study. Chapter 4 presents the analysis of the data generated in the previous chapter. Chapter 5 states the conclusions that have been drawn based on the results of the research and indicates areas for future research. The appendices show samples of the data used in this research and some of the results obtained.

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CHAPTER 2 LITERATURE REVIEW This chapter aims to provide a better understanding of linear project scheduling techniques in the construction industry and reviews some of them. This chapter is composed of four parts: Linear Scheduling Method, Line of Balance Method, stochastic approach of construction scheduling, and probabilistic approach of construction scheduling. Linear Scheduling Method (LSM) The Linear Schedule of a project is the production rate of each of the repetitive activities plotted in a time and location chart. The linear schedule provides a tool to supervise the progress of production crews as they move along the linear aspect of the project. Research Efforts for Linear Scheduling Method Since the early 1960’s, there have been different techniques to schedule linear projects, but for the most part, these have been overshadowed by the Critical Path Method (CPM). The effective planning, scheduling and control of construction projects is necessary. The benefits of implementing and maintaining this set of three management systems are reduced construction time, reduced cost overruns and the minimization of disputes (Callahan et al. 1992). These benefits accrue to the contractor, owner, suppliers and workers in the form of improvements in productivity, quality and resource utilization (Mattila and Abraham 1998). 3

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4 The need for better scheduling techniques is recognized when linear construction projects are considered (Herbsman 1987; Vorster et al. 1992). Linear construction projects consist of a set of activities that are repeated in each location for the length of the job. After an activity is started and/or completed in one location, it has to be repeated in another location. Typically linear construction projects are roadways, tunnels and pipelines. High-rise construction is sometimes classified as linear construction because of the repetitive nature of the work (Callahan et al. 1992; Halpin and Riggs 1992). The time versus distance diagram was an earlier representation of the LSM (Gorman 1972). The reason given for the use of the technique is the better presentation of information. The slope of the plotted line is the production rate of an activity. A linear schedule with time on the horizontal axis and location on the vertical axis was presented with activities represented by line and the slope representing the production rate (Peer 1974). Three different representations are shown in Figure 1-1. Critical activities are those, which have the slowest production rate. Other critical activities are those, which determine the start of the slowest activity or are required to complete the project. The impact of resources on production rates, and therefore, on critical activities was mentioned by Peer (1974), who concluded that resource allocation techniques ignore “the possibility of varying crew size at different stages of the project” (Mattila and Abraham 1998). “Linear balance charts” can be used on linear or repetitive operations (Barrie & Paulson 1978). A pipeline was used to illustrate the concept with time on x-axis and the progress on the y-axis. Activities were plotted as sloping lines that represented production rate. A conflict between two activities was illustrated when lines crossed.

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5 Figure 2-1. Linear schedule (Source: Peer, 1974) A linear scheduling diagram was presented by Clough and Sears (1979) using a pipeline example. This ‘bar chart for repetitive operations’ diagram had location plotted

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6 on the x-axis and time plotted on y-axis. Early start and early finish dates were plotted for activities on the y-axis and the line connecting them was the production rate. A graphic method was used by Stradal and Cacha (1982) called ‘time space scheduling’ to schedule linear projects. Essentially, an activity on a project has a location on a graph in time and location, which once drawn make the interaction with other activities apparent. The horizontal axis represents time and the vertical axis represents location. Claimed advantages include the planning of work in sections, the arrangement of flow lines to provide smooth use of capacities, learning effects and easy in understanding the diagram. Charzanowski and Johnson (1986) presented an application of LSM to a road project. The study stated that the use of LSM should be used as a complement to CPM (Critical Path Method) and can be employed by itself on simple, repetitive projects. From the LSM diagram the resource requirements can be determined and then entered into a table where cumulative quantities can easily be found. Vorster and Parvin (1990) used a highway example to illustrate the LSM (Linear Scheduling Method) procedure and to illustrate the graphic symbols (bars, lines and blocks) used in LSM. A critical path through a LSM (Linear Scheduling Method) was suggested, but no method to calculate it was presented. Vorster and Parvin (1990) described the three factors needed to successfully plan a linear construction job: (1) crews must be given time and space to do their job; (2) work must be performed in an ordered sequence; (3) delays and changes must be minimized. Bafna (1991) stated that better communication between all parties in the construction process can be obtained by using linear scheduling. They saw a need for

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7 tabular data and they also introduced the crew movement chart (CMC). The CMC (Crew Movement Chart) provides information for an activity, including which crew is assigned, what that crew is working on, its location and the date. The importance of visual scheduling and the use of truly graphic linear scheduling were described by Vorster et al. (1992). The difference between truly linear projects and repetitive projects was explained and the use of different applications for each was encouraged. An algorithm was developed by Harmelink (1995). LSMh determines controlling activities and then calculates the controlling path through a LSM schedule. Controlling activities are those activities that, if delayed, delay the completion of the project. The concept of rate float, which is the amount of production rate that a non-controlling segment of an activity can be lowered to, before that activity becomes controlling, was introduced. Other Approaches to Linear Scheduling In his approach, Selinger (1980) introduced a method based on the labor requirement and feasible crew size, rather than “on activity durations determined in advance.” He stated that this method “allows for continuity between activities” and “it is more suitable for the practical needs of a construction site.” One example was used to sustain his findings: a simplified bridge-construction project. The optimum solution is shown in Figure 1-2. Perera (1982) presented a linear programming approach for resource sharing in linear construction. The actual amount of resource-hour needed to complete a task was used rather than assigning a complete resource for a day. This provided more flexibility in the assignment of resources and allowed an opportunity for resources to be shared

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8 between activities. Linear programming was then used to determine the maximum rate of production that could be obtained and the resource-hour required by the activities. The end result was that all the activities proceeded at the same rate. Figure 2-2. Optimum linear schedule for a bridge construction project (Source: Selinger, 1980) Russell and Caselton (1988) used dynamic programming to minimize the project duration of a linear project. The variables considered were the duration of activities and possible interruptions of those activities. The items considered in the model formulation were work continuity, learning curve effects, interruptions of activities and general precedence relationships. A sensitivity analysis, where near optimal solutions are available to the user, permitted schedule alternatives. Huang and Halpin (1994) presented resource utilization using linear programming with the addition of a graphical interface. Fundamentally, this was the same as the Perera (1982) paper, where the optimum resource usage for the activities was found. The graphic input and output provided ease of use in generating the linear programming

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9 problem, and the graphic output made interpretation much easier, especially for the novice. Thabet and Beliveau (1994) presented a method of work-space scheduling for repetitive construction and for multi-storey buildings in particular. The resources that were considered for scheduling were the physical space requirements for material storage, and the movement of the manpower and equipment. The scheduling actions proposed to allocate the space resource were: (1) the adjustment of productivity rates; (2) the interruption of the flow of the activity; (3) the delay in the start of the activity. Dubey (1993) and Russell and Dubey (1995) used the minimum moment algorithm, as described in Harris (1978) to do resource leveling on linear projects. Modifications to Harris’ (1978) algorithm were made to incorporate resource leveling into REPCON (Russel and Wong 1993). These modifications included: the ability to employ activities with variable resource users; the ability to shift activities with multiple locations as a unit. The resulting algorithm was called the modified minimum momentum algorithm (Dubey 1993). Comparison of LSM and CPM Due to an increasingly competitive environment, construction companies are becoming more sophisticated, narrowing their focus, and becoming specialists in certain types of construction. This specialization requires more focused scheduling tools that prove to be better for certain types of projects. There are different kinds and varieties of scheduling tools. These tools vary depending on how they represent and analyze activities and their logical relationships. Some of these have been adapted from manufacturing settings – line of balance (LOB) and are used extensively in the construction industry. Other methods have been used to

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10 schedule train arrivals/departures – linear schedules and were also adapted to be used in construction. The most known and utilized methods in the construction industry are: the Network scheduling type, such as the Critical Path Method (CPM) and the Project Evaluation and Review Technique (PERT); Bar/Gantt chart; Line of Balance (LOB), adapted to construction as the vertical production method (VPM); Linear Schedules (LS) adapted by several methods among which is the Linear Scheduling Method (LSM) (Yamin and Harmelink 2001). Some of these methods are more efficient than others, depending on the nature of the project to be scheduled. Table 2-1 suggests which scheduling tool is appropriate for each type of project (Yamin and Harmelink 2001). LSM (Harmelink 1995) is a specialized tool that improves linear scheduling by allowing CPM -type calculations. The LSM performs optimally when scheduling linear continuous projects, such as highway construction. However, LSM can be very inefficient when scheduling complex discrete projects (bridges, buildings, etc.). The CPM is quite the opposite; it is ineffective and cumbersome for scheduling linear continuous projects but extremely efficient for more complex and discrete type projects. Examples LSM/CPM Comparison of the LSM and the CPM (Yamin and Harmelink 2001) was based on how these two methods can incorporate different project management attributes. Two examples were used to illustrate some of the most notable differences between these methods (a bridge project and a small highway rehabilitation project). These two different types of projects were selected because the first one is discrete and non-continuous and the second is linear and continuous.

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11 Table 2-1. Linear scheduling—Methods in the construction industry Type of project Scheduling method Main characteristic Linear and continuous projects (pipelines, railroads, tunnels, highways) LSM • Few activities Multiunit repetitive projects (housing complex, buildings) LOB • Final product a group of similar units High-rise buildings LOB, VPM • Repetitive activities • Hard logic for some activities, soft for others • Large amount of activities • Every floor considered a production unit Refineries and other very complex projects PERT/CPM • Extremely large number of activities • Complex design • Activities discrete in nature • Crucial to keep project in critical path Simple projects (of any kind) Bar/Gantt chart • Indicates only time dimension (when to start and end activities) • Relatively few activities Source: Yamin and Harmelink (2001) For the first case, a small concrete bridge was scheduled using both CPM and LSM. Only eight activities were used: (1) North foundation (excavating, forming, and placing concrete for the abutment’s foundations); (2) South foundation (same as north foundation); (3) North concrete abutment construction (scaffold erection, rebar assembly, concrete placing, and curing); (4) South concrete abutment construction (same as north abutment); (5) Placement of east concrete beams (beams of 20 m lengths will span from the north abutment toward the south abutment, and the direction north-south is called east abutment); (6) Placement of west concrete beams (same as east concrete beams); (7)

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12 Placement of prefab concrete forms over beams; (8) Pavement (hand poured concrete pavement over prefabricated slabs). From the planning and analysis standpoint, both methods allow the representation of the interrelation of activities in a clear and understandable format (Yamin and Harmelink 2001). However, for activities that occupy the same location at the same time, the LSM representation is somewhat confusing. Yamin and Harmelink (2001) stated that the overlapping of activities is unacceptable, because the LSM method considers that such intersection of activities represents a conflict. Nevertheless, the LSM offers an intuitive visual representation of the sequence in which the activities will perform, as well as the location they will occupy at specific times. Since many of the activities occupy the given amount of space for a certain time, most of the activities are discrete blocks. This reduces the advantages offered from the LSM. Both methods allow the identification of those activities that are critical for the on-time completion of the project and which have float. For the second case, a one lane road that needs to be repaved was considered. For this example the researchers used only three activities: (1) remove old pavement, (2) sub-base replacement and leveling, and (3) pave. Yamin and Harmelink (2001) stated that in terms of project planning and analysis for linear projects, the LSM schedule accurately represents the inherent space-time relationship of the project. To moderately approximate this detail with CPM, activities have to be segmented creating further representation and calculation complications. The conclusions of this study were that “with the use of two simple examples, it was shown that planning is facilitated using LSM, because it is visual and intuitive” (Yamin and Harmelink, 2001). Also, LSM is superior to CPM for very

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13 specific projects (linear and continuous), but CPM is a more complete scheduling tool than LSM, mainly because multiple resource management techniques and statistical analysis have been developed for it. Line of Balance (LOB) Line of Balance (LOB) is a variation of linear scheduling methods that allows the balancing of operations such that each activity is continuously performed, for discrete repeated tasks. The major benefit of the LOB methodology is that it provides production rate and duration information in the form of an easily interpreted graphics format (Arditi and Albulak 1986). Highway construction, housing projects, long bridges, and many other types of construction projects are characterized by repetitive operations. Linear scheduling techniques are known to be the most suitable method for the overall management of such type of construction projects. The Line of Balance technique is one of these linear scheduling methods using known scheduling methods such as the Critical Path Method (CPM), Program Evaluation and Review Technique (PERT), and bar chart, and it does not replace them (Sarraj, 1990). The Line of Balance Method (LOB) was originated by the Goodyear Company in early 1940s, and ten years later U.S. Navy is known to have used a technique called “Line of Balance” to plan and monitor the progress of industrial processes (Department of Navy, 1962). In April of 1962, the office of Naval Material under the Department of the United States Navy described the Line of Balance Technology as follows: “Line of Balance is a technique for assembling, interpreting and presenting in a graphic form the essential factors involved in a production process from raw materials to completion of the end product, against a background of time. It is essentially a management-type tool,

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14 utilizing the principle of execution to show only the most important components into manufacture of end items in accordance with phased delivery requirements.” Several practitioners have applied the Line of Balance method in the construction industry as a scheduling technique. Lamsden (1968) modified the basic Line of Balance technique and applied it to house construction scheduling. He explained in detail the principles of the method, its representation, and the analysis. The objective of this method was to determine or evaluate the flow rate of finished products in a production line (Sarraj, 1990). The Line of Balance technique has three components output: (1) a unit network which shows activity dependences and time required between activity and unit completion; (2) an objective chart showing cumulative calendar schedule of unit completion; (3) a progress chart showing the completion of the activities of each unit (Carr and Meyer, 1974). The unit network shows the assembly operations for a single unit of many to be produced (Johnson, 1981). It also uses time for the horizontal axis and some measure of production on the vertical axis. The objective of the Line of Balance unit network is to schedule or record the cumulative events in the completion of a single unit (Rowings and Harmelink, 1995). This approach might be used in planning the construction of a multi-story building; for example, the single unit may be a hotel room or an entire floor. The progress diagram in the Line of Balance technique is prepared as a bar chart. The bar represents the units produced on a particular day, which is compared to the Line of Balance to determine if the process is on, behind or ahead of schedule (Department of Navy, 1962).

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15 Carr and Meyer (1974) concluded that the Line of Balance method could be useful in the construction of the repetitive building units. In their review, Arditi and Albulak (1986) identified several failings of Line of Balance when applied to construction. In particular, they point out that extreme care must be taken in the estimation of production rates, as the method is sensitive to errors in the activity duration estimated. They also recommended careful selection of the drawing scale and using multiple colors to improve legibility of the schedule. In 1990, Sarraj developed and presented an algorithm that resulted in a mathematical Line of Balance model. According to Sarraj, using this method in its mathematical form enabled the development of production and delivery schedules without drawing a diagram; the graphical representation was merely used for illustrative purposes in the process of project control (Sarraj, 1990). An early attempt to develop a computer application was made to schedule repetitive-unit construction by Arditi and Psarros (1987). It was limited to solving the basic LOB (Line of Balance) problem and was not designed to deal with many implementation-related problems that were later identified. Since Arditi and Psarros’ study in 1987, there have been several attempts to solve the various problems associated whit linear scheduling. Wang and Huang (1998) introduced the multistage linear scheduling (MSL) method based on the concept of a multistage decision process. Hegazy et al. (1993) presented an effort to enhance the capabilities of linear scheduling techniques, making them more practical and more attractive for use in construction. Thabet and Beliveau (1994) described a structured procedure to incorporate vertical and horizontal constrains to schedule repetitive work in

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16 a multi-story building project. Lutz et al. (1994) modeled the impact of learning in their program. Moselhi and El-Rayes (1993) studied cost optimization in association with linear scheduling. Sneouci and Eldin (1996) presented a dynamic programming approach for the scheduling of non-serial linear projects with multiple non-overlapping loop structures. Harmelink and Rowings (1998) developed a linear scheduling method that provides a level of analytical capability to the linear scheduling process. The challenges associated with LOB (Line of Balance) scheduling includes developing an algorithm that handles project acceleration efficiently and accurately, recognizing time and space dependencies, calculating LOB quantities, dealing with resource and milestone constrains, incorporating the occasional nonlinear and discrete activities, defining a radically new concept of criticalness, including the effect of the learning curve, developing an optimal strategy to reduce project duration by increasing the rate of production of selected activities, performing cost optimization, and improving the visual presentation of LOB diagrams (Arditi et al. 2002). Arditi et al. (2002) identified several issues associated with LOB applications and some approaches to address these issues have been generated: (1) a new approach was proposed that allows for the handling of logical and strategic limitations associated with the characteristics of repetitive activities; (2) learning rate of each activity was proposed to be established and then converted in man hour estimates; (3) an algorithm that performs project acceleration was proposed that can help in optimizing total project cost; (4) an approach was proposed that allows nonlinear and discrete activities to be handled by an LOB scheduling system without disrupting the underlying philosophy; (5) various alternatives were proposed, including generating LOB diagrams of individual paths and

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17 converting LOB information into bar chart, that can help improve visual problems of presentation if the system is to be widely accepted. Disadvantages of Line of Balance (LOB) and Critical Path Method (CPM) The disadvantages of the LOB in projects scheduling were outlined by many researchers. Kvananah (1985) indicated that the LOB techniques were designed to model simple repetitive production processes and, therefore, do not transplant readily into a complex and capricious construction environment. Color graphics were recommended by Arditi and Albulak (1986) in order to improve the visualization of LOB. Neale and Raju (1988) tried to refine the LOB in a spreadsheet format by introducing activities that run concurrently. They confronted the complex relationships that their spreadsheet had to express and concluded that it was practically meaningless to draw the output in the form of a diagram with an incomprehensible mass of flow lines. Neale and Neale (1989) stated that LOB can show clearly only a limited amount of information and a limited degree of complexity, especially when using the technique to monitor progress. On the other hand, both practitioners and researchers voiced their disappointment with the CPM application on repetitive projects. One of the main reasons was the vulnerability of CPM to sequence changes of work between the repetitive typical units, which is, on repetitive projects, a matter of choice and strategy and frequently depends on unforeseen circumstances (Rahbar and Rowing 1992). In CPM, there is pure logic between repetitive typical activities and progressing apart from that sequence is considered an error called out-of-sequence progress (Suhail 1993). Other shortcoming of CPM was stated by Pilcher (1976) who indicated that updating networks of repetitive projects presents many problems of organization. Carr and Meyer (1974) in criticizing CPM on this point affirmed, “The order of progress through such a project is usually

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18 chosen by management rather than required by the dependencies of the activities.” Clough and Sears (1991) indicated that the practical application of CPM in repetitive operations was complicated, nevertheless CPM was occasionally used. Also, CPM is not oriented towards providing work continuity for the squads of the repetitive activities, which is fundamental in repetitive construction (Suhail and Neale 1994). Selinger (1980) indicated that CPM, when used as a model of repetitive construction, had ruled out the requirement of creating work continuity to obtain maximum resource utilization. Rahbar and Rowing (1992) stated that CPM is unable to distinguish rates of progress of activities and that the number of units that can be completed within any period of duration is not clearly visible. Attempts of Combining CPM and LOB After the weaknesses of scheduling repetitive construction projects using LOB or CPM were identified, researchers tried to combine the two techniques or sometimes they tried to combine the techniques merits. This reinforced the notion that CPM and LOB are complementary (Carr and Meyer 1974). Schoderbek and Digman (1967) presented program and evaluation review/line of balance technique. Their objective was combining Program Evaluation and Review Technique (PERT) and Line of Balance (LOB) in a system applicable wherever and whenever PERT and LOB can be of service. This was an attempt to integrating linear problems with uncertainty. However, the algorithm of scheduling repetitive activities was not shown. Also it could not be inferred that it was resource driven. Perera’s (1982) method addresses multiple networks within a project, it is resource oriented in computing the duration of the project, and accounts for the float time.

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19 However, because the method uses linear programming, it is sophisticated and can be complex in a real life project application. Rehbar and Rowing (1992) presented a method that is insensitive to out-of-sequence progress, yet detailed to be meaningful to the end user. They basically kept the CPM at the summary level of the discrete non-repetitive activities to avoid a complicated network with out-of-space progress errors, and then switched to implement the LOB at the level of the repetitive activities. Suhail and Neale (1994) stated that their CPM/LOB method “finally revived the LOB and achieved the objective of integrating the merits of the two methods. It centers around resource leveling and the utilization of float times to streamline the scheduling process and achieve the project goals in productivity and reduced costs.” The authors indicated that CPM/LOB method is based on resource-driven rates of completions that can be promptly revised and can produce enhanced LOB information incorporating float times by utilizing the resource management capacity of the CPM software. It eliminates the troubles associated with the change of sequence of operations with a minimal input and indicates their impact as a reality on the completion date. Other achievement of this method is that maintains a practical continuity of work for the squads and respects the branching logical relationships of the typical unit networks (Suhail and Neale 1994). Stochastic/Probabilistic Approach in Project Scheduling With the advent of more powerful and cheaper computing power, particularly in the form of personal computers, it is now feasible to use stochastic simulation more efficiently as a technique to aid the determination of specific economic resource combinations in order to undertake construction activities. It will allow a realistic evaluation of different methods and technologies that may be undertaken in different

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20 circumstances. The simulation studies can be undertaken with a number of objectives in mind, for example, minimizing cost, duration or capital investment, maximizing profit, rate of return or plant utilization, or producing an optimal design. The simulation studies can be used to assist with the process of design, estimating, planning, programming and controlling. The analysis of the sensitivity of a project to changes in circumstances between the point of tendering and actual construction can be undertaken readily and in the widest sense (Pilcher and Flood 1984). Stochastic models allow for uncertainty in the values input to and output from the model. In the Monte Carlo method (for stochastic CPM), it is possible to allow for a range of possible durations for each activity (this range will reflect that observed on site), and the output from the model will include a probability distribution for the duration of the project, a probability distribution for the length of floats, probabilities of activities being critical, and probability distributions for the timing of events. Monte Carlo Method is a rigorous method for assessing the model output uncertainties via the calculation of the output probability distribution function (Shorter and Rabitz 1997), and is not used for linear projects specifically. Dressler (1974) used the time-velocity diagram in the scheduling of linear construction projects. He used on x-axis time and the determined route on y-axis. He stated that “the methodology presented describes the development of a tool to implement decision making in scheduling of linear construction sites on a systematic basis and simultaneously consider the realistic needs of a construction manager. Decision alternatives can be developed by a mathematical model that takes in consideration

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21 stochastic production velocities.” However, “the model cannot by itself provide optimal strategies.” Construction operations involve many uncertain variables and require the use of risk analyses, which experts employ for construction scheduling. To perform these risk analyses, experts use networks to represent the occurrence of activities involved in the construction project. A network consists of activities and links. Each activity represents a significant and definable task in the construction project, while links are used to indicate the relationships between tasks. A sequence of activities that starts with the first activity and ends with the last one is called a path. Failure to complete the project on time occurs when one or more paths take longer to complete than expected (Diaz and Hadipriono, 1993). Diaz and Hadipriono (1993) conducted a study using probabilistic scheduling methods for risk analyses. The methods included in their research were: (1) Program evaluation review technique (PERT); (2) probabilistic network evaluation technique (PNET); (3) narrow reliability bounds (NRB); (4) Monte Carlo simulation (MCS); and (5) simplified Monte Carlo simulation (SMCS). In order to carry out the study, the researchers used 31 construction network cases, the survival function – S(T)=1-F(T), where F(T)= the cumulative distribution function of project duration was used to compare the results of each method, and computer time was measured for each of the five methods. The findings were that “PERT is the simplest method and consistently derives a liberal probability of failure for a network, PNET yields values of probability of having project duration larger than the goal project duration grater than or equal to PERT in every case, NRB provides lower and upper bounds for the probability of the failure” and

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22 “the difference in results obtained by the two simulation methods, MCS and SMCS, and the other three methods, PERT, PNET, NRB, is more evident when the activities in the network have positively skewed distributions” (Diaz and Hadipriono, 1993). All the literature available now shows a lack of stochastic approach in linear projects scheduling and for this reason a study that reveal the impact of uncertainty on construction project performance using linear scheduling is needed.

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CHAPTER 3 RESEARCH OBJECTIVES AND METHODOLOGY Objectives Linear scheduling is a way to satisfy the need for simplicity without sacrificing the wealth of information that needs to be presented. Linear Scheduling provides the planner (manager) with an important tool: the ability to visualize a project. Linear Scheduling provides a simple diagram which shows the location and time at which a given crew will be performing a task. Variation of activities duration is common place in the construction industry. This is due to the fact that the construction industry is highly influenced by variations in weather, productivity, quality of materials, etc. Stochastic analysis has been used to model variations in activities and produce more effective and reliable project duration estimates. Throughout this study, the following restrictions will be considered: construction production rate for each activity will be constant, a single crew will be used for each activity, all work will progress in the same direction, and a buffer of one day will be considered between activities. The objectives of this study were to determine how different degrees of uncertainty impact the construction project duration, how uncertainty affect the activity idle time and the project idle time, and whether the project schedule produce by the deterministic approach have the potential to be more optimum when uncertainty was considered. 23

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24 Multiple simulations have been performed and different parameters have been analyzed. Comparisons between deterministic parameters of the linear scheduling and the outputs from a stochastic approach in linear scheduling have been completed. The methodology used to complete this research study includes two major phases, Generating Data phase and Data Analysis phase, as detailed in the following subsections. Methodology: Generating Data In order to accomplish the objective mentioned above, a typical linear schedule has been selected. The information needed for this research was collected from the article ‘Resource leveling of linear schedules using integer linear programming’, by Mattila and Abraham (1998). In the article the major activities and their production rates (measured in linear foot per day) are presented for the widening of a segment of U.S. Route 41, located in northern Michigan. The segment is 5000 feet long and the only data used in this research where the production rates. To complete this research, a few computer programs have been used. Palisade BestFit software was used in order to fit the available data in statistical distributions and display them in high-resolution graphs. BestFit helped with a fitted probability distribution that gave the best possible modeling results by accurately describing the process. Other modeling computer program used was Pelisade Monte Carlo@Risk. Monte Carlo@Risk software uses probability distributions to describe uncertain values and to present the results, using Excel spreadsheets. This program was used to introduce variances (uncertainty) to the activities of the linear schedule and helped to determine outcomes and probabilities of occurrence.

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25 Throughout this research, several restrictions have been considered: the production rates where considered constant for entire duration of each activity; each activity was performed by only one crew; a buffer between activities of one day was imposed. Deterministic Project Duration BestFit software was used to determine the statistical distribution for the activity production rates for the project (U.S. Route 41). This operation was performed in order to generate thirty different project values that will be analyzed later in this study. Actual production rates where used as input and a graph representing the fit probability distribution was generated. Table 3-1 contains the values of activity production rates (measured in feet per day) and Figure 3-1 shows this distribution and his parameters. Table 3-1. Activity production rates Activity Production Rate Activity 1 333 Activity 2 385 Activity 3 625 Activity 4 1000 Activity 5 1000 Activity 6 1000 Activity 7 1667 Several fit distributions where generated by BestFit. LogLogistic distribution was choose based on the P-P (Probability-Probability) output. This output compares the p-values of the fitted distribution to the p-value of the fitted result and helps in choosing the correct distribution. LogLogistic(,,) distribution is characterized by three parameters: (the location parameter to the distribution), (shape parameter to the distribution and must be greater than zero at any time), and (shape parameter to the distribution and must be greater than zero at any time).

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26 Figure 3-1. LogLogistic distribution. Based on LogLogistic(268.62, 443.71, 1.6293) distribution and the parameters obtained, ten activity production rates where generated using Monte Carlo@Risc computer program. Using these ten activities, a new linear schedule was created. Monte Carlo@Risk is a computer program that extends the analytical capabilities of Microsoft Excel to include risk (uncertainty) analysis and simulation. Risk analysis identifies the range of possible outcomes that can be expected for a spreadsheet result and their relative likelihood of occurrence. The next step in processing data for this research was to run thirty simulations using the function RiskLogLogistic available in Monte Carlo@Risk software. These simulations create thirty sets of values and were considered thirty different construction

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27 projects. These new projects (Project #1 to Project #30) will be used to present the impact of uncertainty on construction project duration. The newly created construction project having ten activities was used. For each simulation one thousand iterations were used. The results of this simulation are presented in Table 3-2 throughout Table3-7. Table 3-2. Production rate values for Project #1 to Project #5 Project #1 #2 #3 #4 #5 Activity 1 327.2545 633.3316 404.5917 2239.743 2238.408 Activity 2 559.9563 787.3782 571.6462 634.6245 776.5641 Activity 3 1521.763 639.8746 664.1128 6783.208 415.852 Activity 4 1031.522 538.7965 544.2313 796.3157 536.0936 Activity 5 685.134 329.4674 751.9929 448.743 1081.697 Activity 6 799.197 1386.448 770.7072 1463.944 686.6055 Activity 7 319.7963 570.0885 1720.834 1590.826 3072.443 Activity 8 918.5577 1032.705 855.2875 477.5502 1038.589 Activity 9 1153.207 1222.864 765.9144 660.7139 1034.663 Activity 10 669.5658 663.2338 586.9739 586.2788 475.9876 Table 3-3. Production rate values for Project #6 to Project #10 Project #6 #7 #8 #9 #10 Activity 1 1915.518 1856.7 724.9503 1205.435 362.454 Activity 2 11634.7 812.9675 4416.546 309.1059 378.4961 Activity 3 700.9951 551.9981 2212.757 1152 2466.602 Activity 4 856.7482 715.4408 690.1013 664.13 365.0595 Activity 5 1287.52 496.6205 532.0197 470.9633 861.2995 Activity 6 676.3842 408.5267 666.8587 386.0552 1370.055 Activity 7 975.2155 878.0814 430.8115 680.2594 354.9659 Activity 8 377.755 570.9402 1460.451 330.9403 564.4744 Activity 9 1570.166 383.4842 308.1525 302.2164 1233.68 Activity 10 2067.561 1085.672 509.4413 467.8676 1040.295

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28 Table 3-4. Production rate values for Project #11 to Project #15 Project #11 #12 #13 #14 #15 Activity 1 422.0347 711.5411 572.4813 326.8321 371.5763 Activity 2 880.7555 1361.353 10026.5 1063.301 394.4604 Activity 3 467.0232 405.3462 381.9234 923.7257 866.3622 Activity 4 659.5101 536.0981 713.7069 478.9324 688.8416 Activity 5 386.4721 446.9643 2279.912 431.8969 839.5675 Activity 6 438.0966 836.6976 698.0154 1015.419 865.3387 Activity 7 4266.625 332.8875 3560.47 936.0354 1210.903 Activity 8 511.2005 1141.818 836.8227 643.2595 544.0093 Activity 9 8554.936 1402.282 751.1862 689.0248 480.8501 Activity 10 21722.6 1698.626 2589.373 637.8556 456.8638 Table 3-5. Production rate values for Project #16 to Project #20 Project #16 #17 #18 #19 #20 Activity 1 377.8477 2027.811 686.5545 837.7566 4388.526 Activity 2 1232.598 661.2865 327.3262 422.526 588.6329 Activity 3 369.2997 1172.403 937.6241 392.2453 1850.586 Activity 4 561.8632 943.2923 579.1937 316.6805 2603.228 Activity 5 1315.448 819.6977 332.9218 2428.378 425.652 Activity 6 704.5181 1192.041 364.4752 1245.747 717.0003 Activity 7 566.3853 757.287 378.34 586.2066 2653.965 Activity 8 536.033 858.3329 580.1232 1464.676 664.5856 Activity 9 761.1858 1694.732 313.5318 432.735 408.3199 Activity 10 374.2154 436.9989 546.7985 677.5234 507.0206 Table 3-6. Production rate values for Project #21 to Project #25 Project #21 #22 #23 #24 #25 Activity 1 1730.011 787.7176 528.6223 506.8647 1541.267 Activity 2 518.7332 456.9987 549.9064 540.3822 313.9977 Activity 3 8951.62 2332.26 500.1218 655.3921 484.4364 Activity 4 1101.016 705.7351 3172.946 1131.659 871.7776 Activity 5 620.4883 456.0057 969.44 854.9691 1093.688 Activity 6 370.168 692.5505 1495.319 535.074 560.3476 Activity 7 2310.989 428.6938 480.8397 838.1509 845.5436 Activity 8 1867.177 445.4982 642.5125 605.1107 928.8484 Activity 9 23040.14 769.9744 2013.117 431.0456 1082.121 Activity 10 975.2489 853.7712 670.0756 586.1664 477.5506

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29 Table 3-7. Production rate values for Project #26 to Project #30 Project #26 #27 #28 #29 #30 Activity 1 439.6215 4906.626 1012.222 1145.39 382.5743 Activity 2 324.1107 1743.013 368.6231 588.4746 536.4295 Activity 3 1127.999 727.1729 1992.453 710.7763 400.9337 Activity 4 707.4678 948.7899 693.4569 316.5247 908.4233 Activity 5 727.3526 1846.156 366.8793 1668.965 427.784 Activity 6 508.0686 961.9917 413.0837 517.7578 916.6462 Activity 7 544.878 470.4795 2020.793 7440.788 714.497 Activity 8 626.934 760.1536 597.8757 514.3003 557.8763 Activity 9 976.0263 711.2379 840.589 994.7442 491.5281 Activity 10 1337.646 1229.97 1377.321 1570.33 2086.611 After the production rates for each project was computed, a linear schedule for each simulation has been performed. For better visualization of the results, a graph was plotted for each trial (see Appendix B). Figure 3-2 shows the linear schedule created for Project #1, and Table 3-8 presents the activity production rates, activity duration, starting time of each activity, and finish time of the activities. The values for all thirty projects are presented in Appendix A. In order to create the linear schedules for all thirty projects, a one day buffer between activities was considered. Also, the production rates where considered constant for the entire duration of each activity, and for each activity only one crew was considered. Activity durations have been calculated knowing that the original production rates used in this research are for a 5000 LF segment of U.S. Route 41. Table 3-8. Production rates, activity duration, start time and finish time for Project #1 Project #1 Production Rate Activity Duration Start Time Finish Time Activity 1 327.2545 15.27863 0.00 15.28 Activity 2 559.9563 8.929269 7.35 16.28 Activity 3 1521.763 3.285663 13.99 17.28 Activity 4 1031.522 4.847207 14.99 19.84 Activity 5 685.134 7.297842 15.99 23.29 Activity 6 799.197 6.256279 18.03 24.29 Activity 7 319.7963 15.63495 19.03 34.67 Activity 8 918.5577 5.443316 30.23 35.67 Activity 9 1153.207 4.335733 32.33 36.67 Activity 10 669.5658 7.467526 33.33 40.80

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30 As can be observed from Figure 3-3, activity buffer has been considered at the beginning of the activities 4, 5, 7, 10, and at the end of the activities 1, 2, 3, 6, 8, 9. Project duration has been plotted on x-axis measured in days, and progress of the project has been plotted on y-axis measured in linear feet. Throughout this research paper, the production rates, activity durations, starting time and finish time for all thirty projects have been considered as deterministic obtained values, and they have been named as deterministic production rates, deterministic activity durations, deterministic starting time and deterministic finish time. In order to achieve the objectives of this research, variances (uncertainties) have been applied to these values. Figure 3-2. Linear schedule for Project #1(deterministic values). In order to achieve the goals of this research, the deterministic values obtained (deterministic starting time, deterministic finish time, deterministic activity durations and deterministic production rates) were considered as primary values.

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31 Activity Variance By introducing uncertainty in activity durations, a better view in how the variance of the activity can possibly influence each activity performed during a construction project have been illustrated. Monte Carlo @Risk was used again to introduce variances for each activity of the project, but this time a normal distribution was used. First, the values for a RiskNormal (0, 10%) function have been generated. In this distribution, the first factor (zero) represents the mean value, and the second factor (10%) represents the standard deviation value. Thirty Monte Carlo @Risk simulations have been run, with 1000 iterations each. Using this process, thirty sets of different activity variances were generated. In order to apply these variances to the deterministic production rates and generate possible new production rates, a RiskNormal distribution was used. In this case, the mean value for the distribution was considered as being the deterministic production rate, and standard deviation was considered the product between the deterministic production rate and the results obtained from RiskNormal (0, 10%) function. Thirty simulations were performed for each of thirty trials separately. Throughout this research paper, these values have been referred to as actual production rates. The formula used for these simulations was: t = RiskNormal [T, T*RiskNormal (0, 10%)] (3-1) Where: t is the actual production rate for an activity “x”. T is the deterministic production rate for activity “x”. This process was repeated two times. First, for a RiskNormal (0, 20%) distribution, and second for a RiskNormal (0, 30%) distribution.

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32 Next, the actual production rates have been plotted into a graph, although using the deterministic start tame of the activities. Figure 3-3 shows an example of how the actual production rates have affected the linear schedule for the deterministic production rate. Table 3-9 shows the actual finish times (delayed finish time). The values used in Figure 3-3 where the values generated for Project #2 by RiskNormal (0, 20%) distribution. As shown in Figure 3-3, by introducing actual production rates, some of the activities became non-linear. Figure 3-3 shows that activity 2, 4, 6, 8, and 9 starts with their actual calculated production rate, but these activities have a faster actual production rate than the previous activity. For this reason, the production rates had to be changed (the finish time of these activities had to be delayed). Table 3-9. Linear schedule values. Trial #2-1 Deterministic start time Actual finish time Activity 1 0.00 7.99 Activity 2 2.54 8.99 Activity 3 3.54 11.43 Activity 4 4.54 12.43 Activity 5 5.54 22.09 Activity 6 18.11 23.09 Activity 7 19.11 29.43 Activity 8 24.04 30.43 Activity 9 25.80 31.43 Activity 10 26.80 37.12 For activity 4, this process is not that obvious in Figure 3-3, and the reason is that the buffer between activity start time 3 and 4 is only one day, and the actual production rate for activity 4 is grater then for activity 3. Only the production rates were changed for activity 1, 3, 5, 7, 10, and these changes affected the rest of the activities.

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33 Linear schedule for actual production rates Trial #2-107.997.082.548.9911.433.5412.434.5422.095.5421.3718.1123.0929.4319.1129.3824.0430.4327.0325.8031.4337.1226.800100020003000400050006000TimeProgress Activity 1 Activity 2 Activity 3 Activity 4 Activity 5 Activity 6 Activity 7 Activity 8 Activity 9 Activity 10 Figure 3-3. Linear schedule (actual production rates). At this point, activity idle time and project idle time of the project have been calculated. Deterministic start time, actual finish time, delayed finish time, and actual production rates have been used to calculate these two parameters. Activity Idle Time Activity idle time represents the difference measured in days between actual finish time of the activity (see Figure 3-3 and Table 3-10) and the not delayed finish time of the activity(see Figure 3-4 and Table 3-10), when the deterministic start time is used. To exemplify the results, Trial #2-1 was used. In Figure 3-4, activities have not been rescheduled, and the conflicts between activities have been kept. The values plotted in Figure 3-4 are shown in Table 3-10. As shown in Table 3-10, the not delayed finish time value for activity 4 is smaller then the not delayed finish time value for activity 3 ( same for activity 9 and 8).

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34 Table 3-10. Linear schedule values (deterministic start time and not delayed finish time) Trial #2-1 Deterministic start time Not delayed finish time Activity 1 0.00 7.99 Activity 2 2.54 8.51 Activity 3 3.54 11.43 Activity 4 4.54 11.26 Activity 5 5.54 22.09 Activity 6 18.11 21.75 Activity 7 19.11 29.43 Activity 8 24.04 29.99 Activity 9 25.80 29.50 Activity 10 26.80 37.12 Rescheduled start time and finish time have been calculated. The findings are shown in Table 3-6 and Figure 3-11. As can be observed, the project duration increased from 37.12 days (not-rescheduled project duration) to 42.06 days (rescheduled project duration). Actual Production Rate Trial #2-118.1119.1124.0425.8026.807.9902.548.5111.433.5411.264.5422.095.5421.7529.4329.9929.5037.120100020003000400050006000TimeProgress Activity 1 Activity 2 Activity 3 Activity 4 Activity 5 Activity 6 Activity 7 Activity 8 Activity 9 Activity 10 Figure 3-4. Linear schedule (not delayed finish time).

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35 Furthermore, if the deterministic project duration (34.33 days) is compared with not-rescheduled project duration, it is observed that the value is smaller. The same conclusion can be made if deterministic project duration is compared with rescheduled project duration. Deterministic project duration for Project #2 can be found in Table 3-12 and Figure 3-6. Table 3-11. Linear schedule values (rescheduled). Trial #2-1 Rescheduled start time Rescheduled finish time Activity 1 0.00 7.99 Activity 2 3.03 8.99 Activity 3 4.03 11.92 Activity 4 6.20 12.92 Activity 5 7.20 23.75 Activity 6 21.11 24.75 Activity 7 22.11 32.43 Activity 8 27.49 33.43 Activity 9 30.73 34.43 Activity 10 31.73 42.06 Actual Production Rate Trial #2-121.1122.1127.4930.7331.737.9903.038.9911.924.0312.926.2023.757.2024.7532.4333.4334.4342.060100020003000400050006000TimeProgress Activity 1 Activity 2 Activity 3 Activity 4 Activity 5 Activity 6 Activity 7 Activity 8 Activity 9 Activity 10 Figure 3-5. Linear schedule (rescheduled).

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36 Table 3-12. Linear schedule values (deterministic) Project #2 Activity Production Rate Start Time Finish Time Activity 1 633.33 0.00 7.89 Activity 2 787.38 2.54 8.89 Activity 3 639.87 3.54 11.36 Activity 4 538.80 4.54 13.82 Activity 5 329.47 5.54 20.72 Activity 6 1386.45 18.11 21.72 Activity 7 570.09 19.11 27.88 Activity 8 1032.71 24.04 28.88 Activity 9 1222.86 25.80 29.88 Activity 10 663.23 26.80 34.33 Project Idle Time Project idle time represents the difference, measured in days, between the start times of two consecutive activities in a linear schedule. This parameter is calculated when the deterministic start time and the actual production rates of the activities are used. In order to illustrate this factor, not-rescheduled linear schedule and rescheduled linear schedule have been used, calculated for Trial #2-6. Figure 3-6. Linear schedule for Project #2 (deterministic values).

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37 As in the activity idle case, not-reschedule project duration was calculated using the deterministic start time and actual production rate of the activities (see Table 3-13). These data have been plotted into a graph presented in Figure 3-7. Figure 3-7 shows that the difference between the start time of activity 2 and the start time of activity 1 is larger then the restriction imposed for this research (one day buffer between activities). In the same time, the difference between the finish times of these two activities is larger then one day buffer. Same observations can be made for activity 6 with regard to activity 5, activity 8 with regard to activity 9, and activity 9 with regard to activity 8. Table 3-13. Linear schedule values (not rescheduled) Trial #2-6 Deterministic start time Not rescheduled finish time Activity 1 0.00 6.99 Activity 2 2.54 8.88 Activity 3 3.54 11.24 Activity 4 4.54 12.24 Activity 5 5.54 18.92 Activity 6 18.11 21.71 Activity 7 19.11 26.94 Activity 8 24.04 29.04 Activity 9 25.80 39.02 Activity 10 26.80 40.02 The rescheduled values for start time and finish time are presented in Table 3-14. These values have been calculated by using actual production rates for Trial #2-6, and the restrictions imposed from the beginning of this research have been kept (one day buffer between activities). The rescheduled linear schedule is presented in Figure 3-8. By comparing the not-rescheduled project duration with the deterministic project duration, it is observed that the first one is bigger than the second. Same statement can be

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38 made when the rescheduled project duration is compared with the deterministic duration, even if the difference is not that large as it is in the first case. Actual Production Rate Trial #2-618.1119.1124.0425.8026.806.9902.548.8811.243.5412.244.5418.925.5421.7126.9429.0439.0240.020100020003000400050006000TimeProgress Activity 1 Activity 2 Activity 3 Activity 4 Activity 5 Activity 6 Activity 7 Activity 8 Activity 9 Activity 10 Figure 3-7. Linear schedule (not-rescheduled) Table 3-14. Linear schedule values (rescheduled). Trial #2-6 Rescheduled start time Rescheduled finish time Activity 1 0.00 6.99 Activity 2 1.65 7.99 Activity 3 2.65 10.35 Activity 4 3.65 11.35 Activity 5 4.65 18.03 Activity 6 15.44 19.03 Activity 7 16.44 24.26 Activity 8 20.26 25.26 Activity 9 21.26 34.49 Activity 10 22.26 35.49 When the rescheduled production duration is compared with the not-rescheduled project duration, the first one is smaller than the second one.

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39 The processes described above have been performed for all thirty projects. The mean values of the project parameters have been calculated. These parameters have been analyzed in order to determine if by introducing uncertainty in a construction project, a more optimum solution then the deterministic solution can be found. Linear schedule for actual production rates Trial #2-606.997.991.6510.352.6511.353.6518.034.6519.0315.4424.2616.4425.2620.2634.4921.2635.4922.260100020003000400050006000TimeProgres Activity 1 Activity 2 Activity 3 Activity 4 Activity 5 Activity 6 Activity 7 Activity 8 Activity 9 Activity 10 Figure 3-8. Linear schedule (rescheduled) Table 3-15 presents the mean parameter values for the variance introduced by a RiskNormal (0, 10%) function, Table 3-16 hold the values for RiskNormal (0, 10%) function, and Table 3-17 presents the values for RiskNormal (0, 30%) function.

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40 Table 3-15. Mean values of the trials for RiskNormal (0, 10%) function. Project A B1 B2 C1 C2 A.I.1 A.I.2 Project 1 40.80 40.81 0.14 40.46 1.99 2.04 2.65 Project 2 34.33 34.33 0.09 35.28 1.98 2.22 1.87 Project 3 28.63 29.43 1.46 29.83 1.88 2.08 1.44 Project 4 35.57 36.15 1.38 35.98 1.46 2.05 1.68 Project 5 32.56 32.51 0.52 33.18 1.78 2.89 2.87 Project 6 29.93 30.76 0.96 30.17 2.31 4.07 4.90 Project 7 30.65 30.70 0.07 30.64 1.01 0.66 0.60 Project 8 41.07 37.89 0.35 31.45 2.30 11.65 2.88 Project 9 42.98 43.59 0.77 43.49 2.39 2.30 1.97 Project 10 45.65 45.72 0.43 46.12 2.29 3.22 3.03 Project 11 39.84 40.06 0.38 40.84 3.49 5.13 5.83 Project 12 35.59 37.21 1.81 36.20 4.07 6.20 7.08 Project 13 40.55 40.88 0.33 40.84 0.98 1.26 0.71 Project 14 34.60 35.42 1.45 35.27 1.61 1.52 1.36 Project 15 30.76 31.03 0.91 31.37 1.47 2.30 1.41 Project 16 44.03 44.11 0.54 44.63 1.16 2.05 1.42 Project 17 29.30 29.29 0.28 29.60 1.36 1.72 1.42 Project 18 41.29 41.76 0.65 41.61 2.55 3.32 2.00 Project 19 39.40 39.49 0.17 39.96 2.04 2.33 2.35 Project 20 37.68 38.36 1.06 38.37 3.17 4.14 4.54 Project 21 37.01 37.24 0.69 37.39 1.49 1.33 1.88 Project 22 33.21 34.74 1.27 34.21 1.53 4.06 3.91 Project 23 34.98 34.98 0.60 35.94 2.60 3.64 6.38 Project 24 29.42 29.99 0.76 30.11 1.72 4.51 4.21 Project 25 35.12 35.77 1.78 36.67 2.97 5.06 6.33 Project 26 30.03 31.41 1.29 31.27 2.29 4.81 2.87 Project 27 24.25 24.94 0.73 24.33 1.07 1.43 1.00 Project 28 39.57 40.20 0.63 39.55 1.57 2.06 1.40 Project 29 41.97 42.43 0.44 42.50 1.99 1.90 1.63 Project 30 36.12 37.10 1.48 37.26 2.27 2.63 1.72 Average 35.90 36.28 0.78 36.15 2.03 3.15 2.78 (For A) Mean STD 5.31

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41 Table 3-16. Mean values of the trials for RiskNormal (0, 20%) Project A B1 B2 C1 C2 A.I.1 A.I.2 Project 1 40.80 41.79 2.29 43.30 5.03 6.50 6.41 Project 2 34.33 36.01 3.94 35.89 5.96 5.97 7.42 Project 3 28.63 31.96 6.67 35.25 7.59 13.47 14.00 Project 4 35.57 39.44 5.85 40.00 5.96 10.10 16.88 Project 5 32.56 33.48 3.96 36.79 6.53 12.43 17.59 Project 6 29.93 33.58 5.52 33.61 5.49 9.05 8.46 Project 7 30.65 40.27 22.33 41.47 23.59 16.85 24.79 Project 8 41.07 51.83 14.94 53.56 15.61 18.11 16.52 Project 9 42.98 52.44 13.79 55.75 16.48 22.99 22.85 Project 10 45.65 48.32 2.42 48.89 5.40 12.56 9.16 Project 11 39.84 42.08 2.84 42.93 4.74 10.32 8.12 Project 12 35.59 39.55 8.13 40.17 14.14 13.48 26.22 Project 13 40.55 43.95 4.48 44.81 9.08 11.95 18.95 Project 14 34.60 39.82 5.41 44.72 12.17 26.55 30.56 Project 15 30.76 33.78 3.46 36.98 4.38 13.11 10.46 Project 16 44.03 50.06 15.88 51.60 16.01 13.02 17.97 Project 17 29.30 31.98 4.76 33.75 6.96 12.20 28.10 Project 18 41.29 46.20 5.45 48.49 8.77 19.59 20.32 Project 19 39.40 43.19 5.85 43.30 8.56 8.78 11.20 Project 20 37.68 41.01 6.17 40.84 9.00 8.89 15.14 Project 21 37.01 39.79 6.14 40.85 9.00 15.02 28.55 Project 22 33.21 36.46 2.82 37.70 5.31 11.03 10.36 Project 23 34.98 35.94 3.08 36.13 4.58 6.18 10.66 Project 24 29.42 33.45 6.45 34.20 6.35 9.58 13.51 Project 25 35.12 36.25 3.99 38.35 5.56 9.98 7.19 Project 26 30.03 37.66 11.05 38.74 12.04 24.17 30.86 Project 27 24.25 26.97 3.61 26.99 4.58 8.84 10.77 Project 28 39.57 44.32 5.40 45.50 7.03 18.03 22.31 Project 29 41.97 45.42 8.53 45.48 11.06 10.73 14.46 Project 30 36.12 43.78 16.16 42.93 17.07 15.63 22.07 Average 35.90 40.03 7.05 41.30 9.13 13.17 16.73 (For A) Mean STD 5.31

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42 Table 3-17. Mean values of the trials for RiskNormal (0, 30%). Project A B1 B2 C1 C2 A.I.1 A.I.2 Project 1 40.80 44.10 7.48 47.49 10.28 38.82 65.99 Project 2 34.33 36.69 4.14 37.44 6.74 13.24 18.02 Project 3 28.63 40.69 33.92 47.90 39.34 116.43 308.98 Project 4 35.57 39.67 5.37 41.70 9.82 18.28 30.74 Project 5 32.56 37.14 7.15 42.14 12.62 20.92 22.68 Project 6 29.93 37.15 7.62 36.69 9.78 20.79 22.15 Project 7 30.65 34.14 8.56 36.44 11.71 19.34 45.61 Project 8 41.07 47.35 18.67 47.40 20.05 7.10 14.40 Project 9 42.98 47.94 5.23 52.61 10.77 22.97 21.88 Project 10 45.65 54.48 7.31 58.94 13.03 42.23 39.51 Project 11 39.84 42.63 5.93 44.90 10.59 27.91 54.10 Project 12 35.59 42.99 21.75 42.97 26.30 27.98 74.78 Project 13 40.55 41.89 1.70 41.02 2.08 3.22 2.69 Project 14 34.60 41.00 13.93 43.48 18.87 30.27 62.17 Project 15 30.76 33.87 4.37 37.02 6.01 16.45 16.07 Project 16 44.03 46.76 7.45 48.94 9.88 17.09 47.33 Project 17 29.30 30.37 5.89 32.46 8.84 17.47 40.06 Project 18 41.29 44.13 4.59 49.50 8.97 27.87 40.93 Project 19 39.40 43.62 5.96 42.75 7.33 6.74 7.35 Project 20 37.68 45.40 20.13 45.80 20.08 14.42 28.17 Project 21 37.01 37.98 3.94 39.83 8.62 12.47 44.19 Project 22 33.21 42.48 8.61 46.37 13.91 35.83 37.29 Project 23 34.98 36.32 3.61 39.53 6.08 15.20 16.22 Project 24 29.42 35.80 12.66 35.59 13.30 8.50 13.07 Project 25 35.12 38.03 7.37 43.58 9.53 39.46 34.76 Project 26 30.03 35.27 3.97 37.47 9.00 27.54 22.52 Project 27 24.25 26.63 4.03 26.21 3.60 3.55 3.80 Project 28 39.57 46.42 9.55 48.64 11.71 25.58 24.67 Project 29 41.97 44.65 4.60 45.38 12.03 12.77 18.43 Project 30 36.12 41.87 6.17 49.37 11.75 43.72 40.14 Average 35.90 40.58 8.72 42.99 12.09 24.47 40.62 (For A) Mean STD 5.31 In the tables 3-15, 3-16, and 3-17, the columns represent the following: Column A: deterministic project duration Column B1: expected project duration-Stochastic (Start not reschedule) (mean) Column B2: standard deviation-Stochastic (Start not reschedule) (mean) Column C1: expected project duration-Stochastic (Start reschedule) (mean)

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43 Column C2: standard deviation-Stochastic (Start reschedule) (mean) Column A.I.1: activity idle expected value (mean) Column A.I.2: standard deviation-activity idle Methodology: Data Analysis Upon completion of the variances of all thirty projects in the data processing phase, all of the data and the results have been analyzed. This was accomplished in three phases: (1) project duration averages analysis, (2) tendencies in the project duration values analysis, and (3) analysis of activity idle and project delays. In the project duration averages phase, the inputs and outputs obtained for this research have been examined for any visible patterns. Analysis of the data can suggest success or failure of the research. During the analysis of the probability of occurrence, the potential outcomes of the research have been discussed. Finally, in the analysis of activity idle and project delays, the tendency of these two parameters have been evaluated in three different situation.

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CHAPTER 4 ANALYSIS OF RESULTS As have been described in Chapter 3, the methodology used for this research was based on data provided by Mattila and Abraham (1998) in their article “Resource leveling of linear schedules using integer linear programming”. These data were the production rates used in widening a segment of U.S. Route 41, located in northern Michigan. In this chapter a sensitivity analysis was done in order to be able to expose how much impact the uncertainty has on construction project performance using linear scheduling. Several simulations have been performed, and different types of possible variances have been applied to the available data. The results of these procedures are shown in Table 3-10, Table 3-11 and Table 3-12. In this chapter, these parameters will be analyzed in three stages: (1) project duration and standard deviation analysis, (2) tendencies in the project duration values analysis, and (3) activity idle and project duration delays analysis. Project Duration and Standard Deviation Project Duration In order to determine a pattern in the evolution of the project duration, the values generated for the project duration with the start of the activity not-rescheduled was compared with the project duration when the start of the activity was rescheduled. The averages of project duration for all thirty projects have been used, and are presented in Table 4-1. The values generated by running RiskNormal (0, 10%), RiskNormal (0, 20%), and 44

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45 Table 4-1. Averages of project duration (days). Function Deterministic Activity start not-rescheduled Activity start rescheduled Project duration RiskNormal(0,10%) 35.90 36.28 36.15 Project duration RiskNormal(0,20%) 35.90 40.03 41.30 Project duration RiskNormal(0,30%) 35.90 40.58 42.99 RiskNormal (0, 30%) functions have been plotted into a graph using the deterministic values of project duration have been used. As shown in Figure 4-1, the values for project duration obtained when only 10% variance has been used, indicates only a small increase in regard with the deterministic project duration. The percentage increase in this case from a deterministic approach to the stochastic one was approximate 1%. This value may appear insignificant, however in cases were big projects are considered, this percentage can have an important impact in terms of time and money. If 20% variance is considered, the percentage increase in project duration jumped to approximate 12%, in the not-rescheduled activity start case, and approximate 15% in case of rescheduled activity start. Finally, when 30% variance was considered, the percentage increase in project duration is located between 13% and 20%. Based on the analysis of these percentages it can be concluded that deterministic project duration can be considered an optimistic value in a construction linear schedule. By using deterministic project duration in a project estimate will introduce errors in the final outcome of the project, in terms of project duration and time of completion for the respective project.

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46 Duration Averages40.5840.0336.2842.9941.335.936.1535363738394041424344450%5%10%15%20%25%30%35%Variance PercentageDuration Start notrescheduled Startrescheduled Figure 4-1. Average project duration. Standard Deviation For all thirty projects, standard deviation values have been calculated in all scenarios that were taken in consideration. The standard deviation represents the deviation of each observation from the arithmetic average of the sample. Table 4-2. Averages of standard deviation (days). Function Deterministic Start not-rescheduled Start rescheduled Variance RiskNormal(0,10%) 5.31 0.78 2.03 Variance RiskNormal(0,20%) 5.31 7.05 9.13 Variance RiskNormal(0,30%) 5.31 8.72 12.09 As can be observed from Figure 4-2, the standard deviation averages values are smaller at 10% variance than for the deterministic values (considered at 0% variance). This phenomenon can be explained by the method that had been used in order to create

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47 these values. In the deterministic values case, only thirty values have been used to calculate the deterministic average standard deviation. When the average standard deviation was calculated for not-reschedule and reschedule values, the mean standard deviation was used for each trial. In addition, to create the thirty deterministic project durations, a RiskLogLogistic function was used. Standard Deviation Averages5.310.788.725.3112.097.059.132.03024681012140%5%10%15%20%25%30%35%Percentage VarianceStandard Deviation Start not rescheduled Start Rescheduled Figure 4-2. Standard deviation average. Figure 4-2 shows that by increasing the level of uncertainty, the standard deviation values increase also. From Figures 4-1 and 4-2 can be observed that project duration averages and standard deviation averages have the same pattern. In both cases, when a bigger variance was applied, the values increased noticeably. Tendencies in the Project Duration Values Three projects have been chosen in order to analyze the evolution of project duration values and variance values (measured in days), when different percentages of

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48 uncertainty have been considered. For deterministic values, 0% uncertainty was considered. As is shown in Table 4-3, for Project #1, the project duration has not increased very much when 10% uncertainty was considered in the case of not-rescheduled value. Furthermore, when the linear schedule was rescheduled, the project duration was found to be smaller. However, when 20% and 30% of uncertainty was considered, the increase in project duration in both cases increased significantly. On the other hand, the variance of the project duration increased continuously for all levels of uncertainty. Table 4-3. Project duration and standard deviation values. Project 0% Uncertainty 10% Uncertainty 20% Uncertainty 30% Uncertainty Parameter Not-rescheduled value Rescheduled value Not-rescheduled value Rescheduled value Not-rescheduled value Rescheduled value Not-rescheduled value Rescheduled value Project #1 Mean value 40.8 40.8 40.81 40.46 41.79 43.3 44.1 47.48 Variance value 0 0 0.14 1.98 2.29 5.96 7.48 10.28 Project #10 Mean value 45.65 45.65 45.72 46.12 48.32 48.89 54.48 58.94 Variance value 0 0 0.43 2.29 2.42 5.4 7.31 13.03 Project #22 Mean value 33.21 33.21 34.74 34.21 36.46 37.7 42.48 46.37 Variance value 0 0 1.27 1.53 2.82 5.31 8.61 13.91

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49 For Project #10, the not-rescheduled and rescheduled project duration values increases when the amount of uncertainty was increased. Similar facts can be observed for the values of the variances. In the case of Project #22, the tendencies of project durations and variances are the same as for Project #10. In addition, for all levels of uncertainty, the rescheduled values for variances are bigger than in the situation when the project was not rescheduled. Probability of Occurrence Probability of occurrence for a specific value of project duration is an important factor in a stochastic approach in linear scheduling. Project #10 has been used in order to illustrate graphically the probability of occurrence for not-rescheduled project values and rescheduled project values. Mean production durations and mean standard deviations were used for Project #10. This trial has been analyzed for 10%, 20%, and 30% uncertainty. In Figure 4-3, Project #10 using 10% uncertainty is presented. A normal distribution has been used in order to generate the graphical representation. The value for the deterministic project duration has also been plotted. The same concepts were used to produce Figure 4-4 and Figure 4-5. These figures represents Project #10 when 20% uncertainty and 30% uncertainty respectively. As can be observed from Figure 4-3, 4-4 and 4-5, when the percentage of the uncertainty was increased, the range of possible values that the project duration can take also increased, in the case of not-rescheduled project as well as for the rescheduled project.

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50 Activity Idle Time and Project Idle Time In a linear schedule, activity idle time or/and project idle time occur when during the completion of a construction project, for some reason, one or more activities change their production rate. Figure 4-3. Probability of occurrence for Project #10 (10% uncertainty) Figure 4-4. Probability of occurrence for Project #10 (20% uncertainty) Activity idle time represents the delay that a task has to take in order not to overlap with the previous activity in the schedule. The project idle time represents the amount of time which an activity can start earlier in a schedule and still not to overlap with the previous tasks. There are several approaches that can be used in order to diminish or completely

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51 Figure 4-5. Probability of occurrence for Project #10 (30% uncertainty) eliminate the activity and project idle time. The one used for this research was to add a delay to the start time for all activities from a project. Each activity was delayed first with 5%, then with 10%, and finally with 15%. Both parameters (activity idle and project idle) have been analyzed in order to reveal if a common tendency can be found. Variation of Activity Idle and Project Idle For these simulations, the not-rescheduled values (deterministic start time) generated for Project #1 have been used. For each case (10% uncertainty, 20% uncertainty, and 30% uncertainty), three sets of values have been generated. The first set was generated for 5% delay in activity start time, the second set was generated for 10% delay in activity start time, and the third one for 15% delay in activity start time. The results of these calculations are shown in Table 4-4. For a better visualization, the values from Table 4-4 have been plotted into three different graphs. Figure 4-6 represents the 10% level of uncertainty, Figure 4-7 shows the 20% level of uncertainty, and Figure 4-8 the 30% level of uncertainty, and they represent the activity idle and project idle variations, when different amount of uncertainty is considered.

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52 Table 4-4. Variation of activity idle and project duration. Project Level of uncertainty Parameter 0% increase in start time 5% increase in start time 10% increase in start time 15% increase in start time Project #1 10% uncertainty Activity idle 2.04 1.32 0.87 0.59 Project idle 0 1.67 3.33 5 20% uncertainty Activity idle 6.5 5.16 4.3 3.76 Project idle 0 1.59 3.21 4.85 30% uncertainty Activity idle 38.82 36.61 34.64 32.84 Project idle 0 1.4 2.84 4.3 Figure 4-6. Activity idle and project idle variation for Project #1(10% uncertainty). A pattern can be observed in Figure 4-6, Figure 4-7, and Figure 4-8. Activity idle has the tendency to decrease when the delay in the start time of the activity is increased. On the other hand, the project idle tends to increase when delay in the start time of the activity is increased. Based on the data just calculated, an analysis of the saves in activity idle and an analysis of the amount of delay in project duration will be conducted and the results will be analyzed.

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53 Figure 4-7. Activity idle and project idle variation for Project #1 (20% uncertainty). Figure 4-8. Activity idle and project idle variation for Project #1 (30% uncertainty). Activity Idle Time Savings The values of the activity idle time savings have been calculated by deducting the value of the activity idle time before the delay was considered from the value of the activity idle time after the delay was considered. Table 4-5 contains these new values. Figure 4-9 shows that the saves activity idle have the tendency to increase when the level of uncertainty is increased. On the other hand, when the amount of the delay in the

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54 start time is increased, the values of activity save decreases. Table 4-5. Saves in activity idle. Project Delay of the start time 10% uncertainty 20% uncertainty 30% uncertainty Project #1 Activity 5% increase in start time 0.72 1.34 20.21 idle 10% increase in start time 1.17 2.2 4.18 saves 15% increase in start time 1.45 2.74 5.98 Saves in activity idle2.214.180.721.341.172.21.452.745.98012345670%10%20%30%40%Percentage of uncertaintyDays 5% increase in Starting Time 10% increase in Starting Time 15% increase in Starting Time Figure 4-9. Saves in activity idle. Amount of Delay in Project Duration The values of the project delays have been calculated by deducting the value of the project delay before an increase in start time was considered from the value of the project delay after an increase in start time was considered. Figure 4-10 is a graphical representation of the results, and Table 4-6 contains these new values.

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55 Table 4-6. Amount of delay in project duration. Project Delay of the start time 0% uncertainty 10% uncertainty 20% uncertainty 30% uncertainty Project #1 Delay 5% increase in start time 2.04 1.67 1.59 1.4 in project 10% increase in start time 4.08 3.33 3.21 2.84 duration 15% increase in start time 6.12 5 4.85 4.3 Delays in Project duration1.42.844.31.672.041.593.334.083.216.1254.85012345670%10%20%30%40%Percentage of uncertainityProject Idle time 5% increase in Starting Time 10% increase in Starting Time 15% increase in Starting Time Figure 4-10. Delays in project duration. As can be observed, the tendency in the values of the project delay is to decrease when the level of uncertainty increase. On the other hand, if the percentage in the start time increases, the value of the delay increases.

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CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS Conclusions Linear scheduling of construction projects represents a major challenge to project managers. These projects require schedules that ensure an uninterrupted and linear progress of the activities. Most linear schedules are developed in a deterministic approach. In these cases, activity durations are given as a single value, usually the most likely duration. Construction projects are complex and present many uncertainties. These uncertainties are not only from the unique nature of the project but also from the diversity of resources and activities. Moreover, external factors have a very significant effect on the outcome of a construction project. These uncertainties can affect two factors that usually determine the success of a project: the schedule of the project (duration) and the budget of the project. This research focused on the impact that uncertainty can have against the project duration, project idle time, activity idle time, and tried to demonstrate that by using a stochastic approach, a more optimum solution for the project duration can be found, and also to test the sensitivity of project performance introduce by the level of uncertainty. Because of the lack of data available, an artificial uncertainty was introduced in this research, and the Monte Carlo @Risk computer program was used to introduce different types and levels of variances to the project activities. 56

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57 The results of these simulations proved that even if the uncertainties are introduced using a generalized method, namely the variance of the uncertainties between activities with 10%, 20% or 30%, a more reliable outcome in terms of project duration have been found. Throughout this research it has been demonstrated that after each level of uncertainty was introduced, the deterministic project duration value was smaller then the average stochastic project duration values obtained. These findings lead to the conclusion that the deterministic project duration in addition to underestimating the project duration, is unlikely to be the optimum schedule in terms of meeting the project time (project duration, project idle time, and project activity idle). In the same time, this research demonstrated that by increasing the level of uncertainty, the value of the project durations is also increases. This demonstrates that uncertainty is a very important factor in linear scheduling. Every time a production rate of an activity is changed due to a variation, two types of idle time may occur. One is the activity idle and the second is the project idle time. In this research, both types of idle time have been analyzed and how these parameters react to changes in the linear schedule have been presented. Again, to do so, an empiric method was used, namely the activities start time have been delayed. The results shown that when the amount of the delay increases, the activity idle values decreases. On the other hand, when the amount of the delay increases, the project idle increases as well, but in construction projects, it is easier to overcome this type of inefficiency. Considering all the obtained results, this research demonstrate that by considering uncertainties when a linear schedule is prepared, managers (planners) will be able to find

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58 a more optimum solution in terms of project duration for any given construction project. Recommendations for Future Research This research only scratches the surface of the impact of uncertainty to construction project duration in linear scheduling. A generalized approach was used in order to obtain the results presented in this paper. However, it demonstrated the importance of taking uncertainty into account in linear projects, and sets the foundation for further more research. An important recommendation for further research would be to conduct the same type of research, but based on actual (real) parameters observed during the completion of a construction project. This will give the researcher the possibility to acknowledge also the solutions adopted in order to reduce the effects of uncertainty. For future researchers will be interesting to look into how the project performance will be affected if the production rate throughout an activity will vary. The next recommendation would be to use different levels of uncertainty observed for a real project for each activity, or only for a number of activities that are considered the most dangerous in terms of variances. This will represent a case study closer to reality than the one presented in this research. An important concern is in how the optimum solution for a construction linear schedule can be found. In this research, the solution proposed was to delay the start time of the activities by a fixed percentage. Different approaches can be used to accomplish this goal, for example to allow for different degrees of delay, or even advancement, for each activity to be introduced. Allocation of extra resources is also an important consideration. The resources can be either man power (extra crew members or extra crews), or equipment. Other research can focus on at which point of the activity progress

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59 the second crew is better to be added, or if the both crews will start at the same time, one at the beginning of the activity and the second at the end of the activity. Also, the learning curve concept can be incorporate in a research, and the variance in the performance of a crew. To incorporate a Genetic Algorithm would be an important asset for this type of research, and would help in manage all the parameters involved.

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APPENDIX A DETERMINISTIC PARAMETERS FOR PROJECT #1 THROUGHOUT PROJECT #30 Table A-1. Production rates, activities duration, start and finish time for Project #1 Project #1 Production Rate Duration Start Time Finish Time Activity 1 327.2545 15.27863 0.00 15.28 Activity 2 559.9563 8.929269 7.35 16.28 Activity 3 1521.763 3.285663 13.99 17.28 Activity 4 1031.522 4.847207 14.99 19.84 Activity 5 685.134 7.297842 15.99 23.29 Activity 6 799.197 6.256279 18.03 24.29 Activity 7 319.7963 15.63495 19.03 34.67 Activity 8 918.5577 5.443316 30.23 35.67 Activity 9 1153.207 4.335733 32.33 36.67 Activity 10 669.5658 7.467526 33.33 40.80 Table A-2. Production rates, activities duration, start and finish time for Project #2 Project #2 Production Rate Duration Start Time Finish Time Activity 1 633.3316 7.894759 0.00 7.89 Activity 2 787.3782 6.350188 2.54 8.89 Activity 3 639.8746 7.81403 3.54 11.36 Activity 4 538.7965 9.279942 4.54 13.82 Activity 5 329.4674 15.17601 5.54 20.72 Activity 6 1386.448 3.606337 18.11 21.72 Activity 7 570.0885 8.770568 19.11 27.88 Activity 8 1032.705 4.841653 24.04 28.88 Activity 9 1222.864 4.088764 25.80 29.88 Activity 10 663.2338 7.538819 26.80 34.33 60

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61 Table A-3. Production rates, activities duration, start and finish time for Project #3 Project #3 Production Rate Duration Start Time Finish Time Activity 1 404.5917 12.35814 0.00 12.36 Activity 2 571.6462 8.746669 4.61 13.36 Activity 3 664.1128 7.528842 6.83 14.36 Activity 4 544.2313 9.18727 7.83 17.02 Activity 5 751.9929 6.648999 11.37 18.02 Activity 6 770.7072 6.487548 12.53 19.02 Activity 7 1720.834 2.905568 17.11 20.02 Activity 8 855.2875 5.845988 18.11 23.96 Activity 9 765.9144 6.528145 19.11 25.64 Activity 10 586.9739 8.518266 20.11 28.63 Table A-4. Production rates, activities duration, start and finish time for Project #4 Project #4 Production Rate Duration Start Time Finish Time Activity 1 2239.743 2.232399 0.00 2.23 Activity 2 634.6245 7.878675 1.00 8.88 Activity 3 6783.208 0.737114 9.14 9.88 Activity 4 796.3157 6.278917 10.14 16.42 Activity 5 448.743 11.14223 11.14 22.28 Activity 6 1463.944 3.415431 19.87 23.28 Activity 7 1590.826 3.143022 21.14 24.28 Activity 8 477.5502 10.4701 22.14 32.61 Activity 9 660.7139 7.567572 26.04 33.61 Activity 10 586.2788 8.528365 27.04 35.57 Table A-5. Production rates, activities duration, start and finish time for Project #5 Project #5 Production Rate Duration Start Time Finish Time Activity 1 2238.408 2.23373 0.00 2.23 Activity 2 776.5641 6.438619 1.00 7.44 Activity 3 415.852 12.02351 2.00 14.02 Activity 4 536.0936 9.32673 5.70 15.02 Activity 5 1081.697 4.622368 11.40 16.02 Activity 6 686.6055 7.282202 12.40 19.68 Activity 7 3072.443 1.627369 19.06 20.68 Activity 8 1038.589 4.814226 20.06 24.87 Activity 9 1034.663 4.832493 21.06 25.89 Activity 10 475.9876 10.50448 22.06 32.56

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62 Table A-6. Production rates, activities duration, start and finish time for Project #6 Project #6 Production Rate Duration Start Time Finish Time Activity 1 1915.518 2.61026 0.00 2.61 Activity 2 11634.7 0.429749 3.18 3.61 Activity 3 700.9951 7.132717 4.18 11.31 Activity 4 856.7482 5.83602 6.48 12.31 Activity 5 1287.52 3.883434 9.43 13.31 Activity 6 676.3842 7.392248 10.43 17.82 Activity 7 975.2155 5.127072 13.69 18.82 Activity 8 377.755 13.23609 14.69 27.93 Activity 9 1570.166 3.184377 25.75 28.93 Activity 10 2067.561 2.418309 27.51 29.93 Table A-7. Production rates, activities duration, start and finish time for Project #7 Project #7 Production Rate Duration Start Time Finish Time Activity 1 1856.7 2.69295 0.00 2.69 Activity 2 812.9675 6.150307 1.00 7.15 Activity 3 551.9981 9.058002 2.00 11.06 Activity 4 715.4408 6.988698 5.07 12.06 Activity 5 496.6205 10.06805 6.07 16.14 Activity 6 408.5267 12.2391 7.07 19.31 Activity 7 878.0814 5.694233 14.61 20.31 Activity 8 570.9402 8.757484 15.61 24.37 Activity 9 383.4842 13.03835 16.61 29.65 Activity 10 1085.672 4.605444 26.05 30.65 Table A-8. Production rates, activities duration, start and finish time for Project #8 Project #8 Production Rate Duration Start Time Finish Time Activity 1 724.9503 6.897025 0.00 6.90 Activity 2 4416.546 1.132106 6.76 7.90 Activity 3 2212.757 2.259624 7.76 10.02 Activity 4 690.1013 7.245313 8.76 16.01 Activity 5 532.0197 9.398148 9.76 19.16 Activity 6 666.8587 7.497841 12.67 20.16 Activity 7 430.8115 11.606 13.67 25.27 Activity 8 1460.451 3.4236 22.85 26.27 Activity 9 308.1525 16.22573 23.85 40.07 Activity 10 509.4413 9.814673 31.26 41.07

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63 Table A-9. Production rates, activities duration, start and finish time for Project #9 Project 9 Production Rate Duration Start Time Finish Time Activity 1 1205.435 4.147879 0.00 4.15 Activity 2 309.1059 16.17568 1.00 17.18 Activity 3 1152 4.340276 13.84 18.18 Activity 4 664.13 7.528646 14.84 22.36 Activity 5 470.9633 10.61654 15.84 26.45 Activity 6 386.0552 12.95151 16.84 29.79 Activity 7 680.2594 7.350137 23.44 30.79 Activity 8 330.9403 15.10846 24.44 39.55 Activity 9 302.2164 16.54444 25.44 41.98 Activity 10 467.8676 10.68678 32.29 42.98 Table A-10. Production rates, activities duration, start and finish time for Project #10 Project #10 Production Rate Duration Start Time Finish Time Activity 1 362.454 13.79486 0.00 13.79 Activity 2 378.4961 13.21018 1.58 14.79 Activity 3 2466.602 2.027081 13.77 15.79 Activity 4 365.0595 13.6964 14.77 28.46 Activity 5 861.2995 5.805181 23.66 29.46 Activity 6 1370.055 3.649489 26.81 30.46 Activity 7 354.9659 14.08586 27.81 41.90 Activity 8 564.4744 8.857797 34.04 42.90 Activity 9 1233.68 4.052915 39.85 43.90 Activity 10 1040.295 4.806328 40.85 45.65 Table A-11. Production rates, activities duration, start and finish time for Project #11 Project #11 Production Rate Duration Start Time Finish Time Activity 1 422.0347 11.84737 0.00 11.85 Activity 2 880.7555 5.676944 7.17 12.85 Activity 3 467.0232 10.70611 8.17 18.88 Activity 4 659.5101 7.581385 12.30 19.88 Activity 5 386.4721 12.93754 13.30 26.23 Activity 6 438.0966 11.41301 15.82 27.23 Activity 7 4266.625 1.171886 27.06 28.23 Activity 8 511.2005 9.780899 28.06 37.84 Activity 9 8554.936 0.584458 38.26 38.84 Activity 10 21722.6 0.230175 39.61 39.84

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64 Table A-12. Production rates, activities duration, start and finish time for Project #12 Project #12 Production Rate Duration Start Time Finish Time Activity 1 711.5411 7.0270010.00 7.03 Activity 10 637.8556 7.83876526.77 34.60 Activity 2 1361.353 3.6728164.35 8.03 Activity 3 405.3462 12.335135.35 17.69 Activity 4 536.0981 9.3266519.36 18.69 Activity 5 446.9643 11.1865810.36 21.55 Activity 6 836.6976 5.97587416.57 22.55 Activity 7 332.8875 15.0200917.57 32.59 Activity 8 1141.818 4.37898129.21 33.59 Activity 9 1402.282 3.56561831.03 34.59 Activity 10 1698.626 2.94355532.65 35.59 Table A-13. Production rates, activities duration, start and finish time for Project #13 Project #13 Production Rate Duration Start Time Finish Time Activity 1 572.4813 8.7339090.00 8.73 Activity 2 10026.5 0.4986799.24 9.73 Activity 3 381.9234 13.0916310.24 23.33 Activity 4 713.7069 7.00567717.32 24.33 Activity 5 2279.912 2.19306723.13 25.33 Activity 6 698.0154 7.16316624.13 31.30 Activity 7 3560.47 1.40430930.89 32.30 Activity 8 836.8227 5.97498131.89 37.87 Activity 9 751.1862 6.65613932.89 39.55 Activity 10 2589.373 1.93096938.62 40.55 Table A-14. Production rates, activities duration, start and finish time for Project #14 Project #14 Production Rate Duration Start Time Finish Time Activity 1 326.8321 15.298380.00 15.30 Activity 2 1063.301 4.70233811.60 16.30 Activity 3 923.7257 5.41286212.60 18.01 Activity 4 478.9324 10.4398913.60 24.04 Activity 5 431.8969 11.5768414.60 26.17 Activity 6 1015.419 4.92407622.25 27.17 Activity 7 936.0354 5.34167923.25 28.59 Activity 8 643.2595 7.77291324.25 32.02 Activity 9 689.0248 7.25663325.77 33.02

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65 Table A-15. Production rates, activities duration, start and finish time for Project #15 Project #15 Production Rate Duration Start Time Finish Time Activity 1 371.5763 13.456180.00 13.46 Activity 10 436.9989 11.4416817.85 29.30 Activity 2 394.4604 12.675541.78 14.46 Activity 3 866.3622 5.7712589.68 15.46 Activity 4 688.8416 7.25856310.68 17.94 Activity 5 839.5675 5.95544712.99 18.94 Activity 6 865.3387 5.77808414.17 19.94 Activity 7 1210.903 4.12915 16.81 20.94 Activity 8 544.0093 9.19101917.81 27.01 Activity 9 480.8501 10.3982518.81 29.21 Activity 10 456.8638 10.9441819.81 30.76 Table A-16. Production rates, activities duration, start and finish time for Project #16 Project #16 Production Rate Duration Start Time Finish Time Activity 1 377.8477 13.232850.00 13.23 Activity 2 1232.598 4.05647410.18 14.23 Activity 3 369.2997 13.5391411.18 24.72 Activity 4 561.8632 8.89896316.82 25.72 Activity 5 1315.448 3.80098622.91 26.72 Activity 6 704.5181 7.09705 23.91 31.01 Activity 7 566.3853 8.82791324.91 33.74 Activity 8 536.033 9.32778425.91 35.24 Activity 9 761.1858 6.56869929.67 36.24 Activity 10 374.2154 13.3612930.67 44.03 Table A-17. Production rates, activities duration, start and finish time for Project #17 Project #17 Production Rate Duration Start Time Finish Time Activity 1 2027.811 2.4657130.00 2.47 Activity 2 661.2865 7.56102 1.00 8.56 Activity 3 1172.403 4.2647475.30 9.56 Activity 4 943.2923 5.3005846.30 11.60 Activity 5 819.6977 6.09981 7.30 13.40 Activity 6 1192.041 4.19448710.20 14.40 Activity 7 757.287 6.60251711.20 17.80 Activity 8 858.3329 5.82524512.98 18.80 Activity 9 1694.732 2.95032 16.85 19.80

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66 Table A-18. Production rates, activities duration, start and finish time for Project #18 Project #18 Production Rate Duration Start Time Finish Time Activity 1 686.5545 7.2827430.00 7.28 Activity 10 507.0206 9.86153327.82 37.68 Activity 2 327.3262 15.275281.00 16.28 Activity 3 937.6241 5.33262711.94 17.28 Activity 4 579.1937 8.63269112.94 21.58 Activity 5 332.9218 15.0185413.94 28.96 Activity 6 364.4752 13.7183516.24 29.96 Activity 7 378.34 13.2156217.75 30.96 Activity 8 580.1232 8.61885923.34 31.96 Activity 9 313.5318 15.9473424.34 40.29 Activity 10 546.7985 9.14413632.15 41.29 Table A-19. Production rates, activities duration, start and finish time for Project #19 Project #19 Production Rate Duration Start Time Finish Time Activity 1 837.7566 5.9683210.00 5.97 Activity 2 422.526 11.833591.00 12.83 Activity 3 392.2453 12.747132.00 14.75 Activity 4 316.6805 15.788783.00 18.79 Activity 5 2428.378 2.05898717.73 19.79 Activity 6 1245.747 4.01365518.73 22.74 Activity 7 586.2066 8.52941719.73 28.26 Activity 8 1464.676 3.41372425.85 29.26 Activity 9 432.735 11.5544126.85 38.40 Activity 10 677.5234 7.37981932.02 39.40 Table A-20. Production rates, activities duration, start and finish time for Project #20 Project #20 Production Rate Duration Start Time Finish Time Activity 1 4388.526 1.1393350.00 1.14 Activity 2 588.6329 8.4942591.00 9.49 Activity 3 1850.586 2.7018477.79 10.49 Activity 4 2603.228 1.9206939.57 11.49 Activity 5 425.652 11.7466810.57 22.32 Activity 6 717.0003 6.97349816.35 23.32 Activity 7 2653.965 1.88397422.44 24.32 Activity 8 664.5856 7.52348523.44 30.96 Activity 9 408.3199 12.2453 24.44 36.68

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67 Table A-21. Production rates, activities duration, start and finish time for Project #21 Project #21 Production Rate Duration Start Time Finish Time Activity 1 1730.011 2.8901550.00 2.89 Activity 10 670.0756 7.46184427.52 34.98 Activity 2 518.7332 9.6388671.00 10.64 Activity 3 8951.62 0.55855811.08 11.64 Activity 4 1101.016 4.54125812.08 16.62 Activity 5 620.4883 8.05816913.08 21.14 Activity 6 370.168 13.5073814.08 27.59 Activity 7 2310.989 2.16357626.42 28.59 Activity 8 1867.177 2.67784 27.42 30.10 Activity 9 23040.14 0.21701330.88 31.10 Activity 10 975.2489 5.12689631.88 37.01 Table A-22. Production rates, activities duration, start and finish time for Project #22 Project #22 Production Rate Duration Start Time Finish Time Activity 1 787.7176 6.3474530.00 6.35 Activity 2 456.9987 10.940951.00 11.94 Activity 3 2332.26 2.14384310.80 12.94 Activity 4 705.7351 7.08481111.80 18.88 Activity 5 456.0057 10.9647812.80 23.76 Activity 6 692.5505 7.21969 17.54 24.76 Activity 7 428.6938 11.6633418.54 30.21 Activity 8 445.4982 11.2233919.98 31.21 Activity 9 769.9744 6.49372325.71 32.21 Activity 10 853.7712 5.85637 27.35 33.21 Table A-23. Production rates, activities duration, start and finish time for Project #23 Project #23 Production Rate Duration Start Time Finish Time Activity 1 528.6223 9.4585490.00 9.46 Activity 2 549.9064 9.0924561.37 10.46 Activity 3 500.1218 9.9975652.37 12.36 Activity 4 3172.946 1.57582311.79 13.36 Activity 5 969.44 5.15761712.79 17.95 Activity 6 1495.319 3.34376815.60 18.95 Activity 7 480.8397 10.3984816.60 27.00 Activity 8 642.5125 7.78194920.22 28.00 Activity 9 2013.117 2.48371126.52 29.00

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68 Table A-24. Production rates, activities duration, start and finish time for Project #24 Project #24 Production Rate Duration Start Time Finish Time Activity 1 506.8647 9.8645650.00 9.86 Activity 10 1337.646 3.73790926.29 30.03 Activity 2 540.3822 9.25271 -8.25 10.86 Activity 3 655.3921 7.629021-6.63 11.86 Activity 4 1131.659 4.418292-3.42 12.86 Activity 5 854.9691 5.8481651.00 15.29 Activity 6 535.074 9.3445021.00 19.79 Activity 7 838.1509 5.965513-4.97 20.79 Activity 8 605.1107 8.2629511.00 24.09 Activity 9 431.0456 11.5997 1.00 28.42 Activity 10 586.1664 8.530001-7.53 29.42 Table A-25. Production rates, activities duration, start and finish time for Project #25 Project #25 Production Rate Duration Start Time Finish Time Activity 1 1541.267 3.2440840.00 3.24 Activity 2 313.9977 15.923681.00 16.92 Activity 3 484.4364 10.321277.60 17.92 Activity 4 871.7776 5.73540813.19 18.92 Activity 5 1093.688 4.57168815.35 19.92 Activity 6 560.3476 8.92303216.35 25.28 Activity 7 845.5436 5.91335620.36 26.28 Activity 8 928.8484 5.38301 21.89 27.28 Activity 9 1082.121 4.62055623.65 28.28 Activity 10 477.5506 10.4700924.65 35.12 Table A-26. Production rates, activities duration, start and finish time for Project #26 Project #26 Production Rate Duration Start Time Finish Time Activity 1 439.6215 11.373420.00 11.37 Activity 2 324.1107 15.426831.00 16.43 Activity 3 1127.999 4.43262912.99 17.43 Activity 4 707.4678 7.06746 13.99 21.06 Activity 5 727.3526 6.87424515.19 22.06 Activity 6 508.0686 9.84119116.19 26.03 Activity 7 544.878 9.17636617.85 27.03 Activity 8 626.934 7.97532120.05 28.03 Activity 9 976.0263 5.12281323.91 29.03

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69 Table A-27. Production rates, activities duration, start and finish time for Project #27 Project #27 Production Rate Duration Start Time Finish Time Activity 1 4906.626 1.01903 0.00 1.02 Activity 10 1570.33 3.18404438.79 41.97 Activity 2 1743.013 2.8685961.00 3.87 Activity 3 727.1729 6.8759442.00 8.88 Activity 4 948.7899 5.26987 4.61 9.88 Activity 5 1846.156 2.70833 8.17 10.88 Activity 6 961.9917 5.19755 9.17 14.37 Activity 7 470.4795 10.6274610.17 20.80 Activity 8 760.1536 6.57761815.22 21.80 Activity 9 711.2379 7.02999716.22 23.25 Activity 10 1229.97 4.06513920.18 24.25 Table A-28. Production rates, activities duration, start and finish time for Project #28 Project #28 Production Rate Duration Start Time Finish Time Activity 1 1012.222 4.9396280.00 4.94 Activity 2 368.6231 13.563991.00 14.56 Activity 3 1992.453 2.50947 13.05 15.56 Activity 4 693.4569 7.21025414.05 21.26 Activity 5 366.8793 13.6284615.05 28.68 Activity 6 413.0837 12.1040917.58 29.68 Activity 7 2020.793 2.47427728.21 30.68 Activity 8 597.8757 8.36294229.21 37.57 Activity 9 840.589 5.94821 32.62 38.57 Activity 10 1377.321 3.63023635.94 39.57 Table A-29. Production rates, activities duration, start and finish time for Project #29 Project #29 Production Rate Duration Start Time Finish Time Activity 1 1145.39 4.3653270.00 4.37 Activity 2 588.4746 8.4965431.00 9.50 Activity 3 710.7763 7.0345623.46 10.50 Activity 4 316.5247 15.796554.46 20.26 Activity 5 1668.965 2.99586818.26 21.26 Activity 6 517.7578 9.65702619.26 28.92 Activity 7 7440.788 0.67197229.25 29.92 Activity 8 514.3003 9.72194730.25 39.97 Activity 9 994.7442 5.02641835.94 40.97

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70 Table A-30. Production rates, activities duration, start and finish time for Project #30 Project #30 Production Rate Duration Start Time Finish Time Activity 1 382.5743 13.069360.00 13.07 Activity 2 536.4295 9.3208894.75 14.07 Activity 3 400.9337 12.470895.75 18.22 Activity 4 908.4233 5.50404213.72 19.22 Activity 5 427.784 11.6881414.72 26.40 Activity 6 916.6462 5.45466721.95 27.40 Activity 7 714.497 6.99793 22.95 29.95 Activity 8 557.8763 8.96256 23.95 32.91 Activity 9 491.5281 10.1723624.95 35.12 Activity 10 2086.611 2.39623 33.72 36.12

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APPENDIX B DETERMINISTIC LINEAR SCHEDULES Figure B-1. Deterministic linear schedule Project #1. 71

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72 Figure B-2. Deterministic linear schedule Project #2 Figure B-3. Deterministic linear schedule Project #3.

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73 Figure B-4. Deterministic linear schedule Project #4. Figure B-5. Deterministic linear schedule Project #5.

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74 Figure B-6. Deterministic linear schedule Project #6. Figure B-7. Deterministic linear schedule Project #7.

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75 Figure B-8. Deterministic linear schedule Project #8. Figure B-9. Deterministic linear schedule Project #9.

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76 Figure B-10. Deterministic linear schedule Project #10. Figure B-11. Deterministic linear schedule Project #11.

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77 Figure B-12. Deterministic linear schedule Project #12. Figure B-13. Deterministic linear schedule Project #13.

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78 Figure B-14. Deterministic linear schedule Project #14. Figure B-15. Deterministic linear schedule Project #15.

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79 Figure B-16. Deterministic linear schedule Project #16. Figure B-17. Deterministic linear schedule Project #17.

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80 Figure B-18. Deterministic linear schedule Project #18. Figure B-19. Deterministic linear schedule Project #19.

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81 Figure B-20. Deterministic linear schedule Project #20. Figure B-21. Deterministic linear schedule Project #21.

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82 Figure B-22. Deterministic linear schedule Project #22. Figure B-23. Deterministic linear schedule Project #23.

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83 Figure B-24. Deterministic linear schedule Project #24. Figure B-25. Deterministic linear schedule Project #25.

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84 Figure B-26. Deterministic linear schedule Project #26. Figure B-27. Deterministic linear schedule Project #27.

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85 Figure B-28. Deterministic linear schedule Project #28. Figure B-29. Deterministic linear schedule Project #29.

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86 Figure B-30. Deterministic linear schedule Project #30.

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LIST OF REFERENCES Arditi, D., and Albulak, M.Z. (1986). Line of balance scheduling in pavement construction, Journal of Construction Engineering and Management, ASCE, 112 (3), 411-424. Arditi, D., and Psarros, M. K. (1987). SYRUS: System for repetitive unit schedule process, NORDNET/INTERNET/PMI: Conf. on Project Management, Vekefnastjomun, the Icelandic Project Management Society, Reykjavik, Iceland. Arditi, D., Tokdemir, O.B., and Suh, K. (2002). Challenges in line of balance scheduling, Journal of Construction Engineering and Management, ASCE, 128 (6), 545-556. Bafna, T. (1991). Extending the range of linear scheduling in highway construction, Master’s Thesis, Department of Civil Engineering, Virginia Polytechnic Institute and State University, Blackburg, VA. Barrie, D. S., and Paulson, B. C. (1992). Professional construction management, McGraw-Hill, Inc., N.Y. Callahan, M.T., Quackenbush, D.G. and Rowings, J.E. (1992). Construction project scheduling, McGraw-Hill, New York, NY. Carr, R.I. and Meyer, W.L. (1974). Planning construction of repetitive building units, Journal of the Construction Division, ASCE, 100 (3), 403-412. Charzanowski, E.N., Jr. and Johnson, D.W. (1986). Application of linear construction, Journal of Construction Engineering, ASCE, 112 (4), 476-491. Clough, R.H. and Sears, G.A. (1979). Construction project management, 2nd Ed., John Wiley & Sons, New York, NY. Clough, R.H. and Sears, G.A. (1991). Construction project management, 3rd Ed., John Wiley & Sons, New York, NY. Department of the Navy, Office of Naval Material (April 1962). Line of balance technology: a graphical method of industrial programming, Navexos P1851, Washington, D.C. Diaz, C.F. and Hadipriono, F.C. (1993). Nondeterministic networking methods, Journal of Construction Engineering and Management, ASCE, 119 (1), 40-57. 87

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88 Dressler, J. (1974). Stochastic scheduling of linear construction sites, Journal of the Construction Division, ASCE, 100 (4), 571-587. Dubey, A. (1993). Resource leveling and linear construction, Master’s thesis, Department of Civil Engineering, University of British Columbia, Vancouver, Canada. Gorman, L.E. (1972). How to get visual impact on planning drawings, Roads and streets, 115 (8), 74-75. Harris, R.B. (1978). Precedence and arrow networking techniques for construction, John Wiley and Sons, Inc., New York, N.Y. Halpin, D.W. and Riggs, L.S. (1992). Planning and analysis of construction operations, John Wiley and Sons, New York, NY. Harmelink, D.J. (1995). Linear scheduling model: the development of a linear scheduling model with micro: computer applications for highway construction control, PhD. Thesis, Iowa State University, Ames, IA. Harmelink, D.J., and Rowings, J.E. (1998). Linear scheduling model: development of controlling activity path, Journal of Construction Engineering and Management, ASCE, 124 (4), 263-268. Hegazy, T., Moselhi, O., and Fazio, P. (1993). BAL: An algorithm for scheduling and control of linear projects, AACE Transactions, Morgantown, W. Va., 8.1-8.14. Herbsman, Z.J. (1987). Evaluation of scheduling techniques for highway construction projects, Transportation Research Record, 1126, 110-120. Huang, R. and Halping, D.W. (1994). Graphically-based LP modeling for linear construction scheduling, Proceedings of Equipment Resource Management into the 21st Century, ASCE, Nashville, TN., (ed. S.L. Weber), 148-155, ASCE, Nashville, TN. Johnson, D.W. (1981). Linear scheduling method for highway construction, Journal of the Construction Division, ASCE, 107 (CO2), 247-261. Kavanagh, D.P. (1985). SIREN: A repetitive construction simulation model, Journal of Construction Engineering and Management, ASCE, 111 (3), 308-323. Lamsden, P. (1968). The line of balance method, Pergamon Press, London, U.K. Lutz, J.D., Halpin, D.W. and Wilson, J.R. (1994). Simulation of learning development in repetitive construction, Journal of Construction Engineering and Management, ASCE, 120 (4), 753-773.

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BIOGRAPHICAL SKETCH Iulian Trofin was born and raised in Bucharest, Romania. He received a Bachelor of Science in Civil Engineering from the University of Civil Engineering of Bucharest in 1993. Iulian worked in the construction field for thirteen years holding different job positions: construction worker, project superintendent, project manager. In 1999, he and his wife came to the United States. In 2001 Iulian was accepted as a graduate student in the Building Construction Department at the University of Florida. In the summer of 2003 he worked as an intern at G.W. Robinson in its estimating department. While in the building construction program, Iulian was a member of Sigma Lamnda Chi (Building Construction Honor Society). His expected graduation date is May 2004, with a Master of Science in Building Construction. 91