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1 Deterministic and Probabilistic Analysis
CA4PRS estimates can be produced from two types of analysis methods: deterministic and probabilistic. Deterministic estimation treats all of the input parameters as constants during productivity calculation, which does not capture the variations frequently seen during construction. In contrast, probabilistic analysis treats all input parameters as variables that change according to an assigned probability distribution function. The probability distribution predicts the likely behavior of an input parameter over a range of potential input parameter values. Probabilistic estimation is the preferred means of CA4PRS analysis because this type of estimation can define and incorporate the uncertainty associated with determining each scheduling or resource input parameter. Probabilistic analysis also yields a more comprehensive estimate than deterministic analysis by providing a range of likely construction productivity, but requires more information about expected variable behavior and the likely variable probability distributions. Included in this documentation is a general description and guide on selecting and using appropriate distribution functions for the different CA4PRS input parameters. Distribution and distribution parameter recommendations are based on data collected from rehabilitation projects on I-15 Devore and I-10 Pomona in California and the I-5 James to Olive Streets Pavement Rehabilitation project completed in Seattle, Washington.
2 Monte Carlo Simulation
In a probabilistic analysis, CA4PRS combines probability distribution functions with Monte Carlo simulation. Monte Carlo simulations refer to a stochastic problem-solving process that is used for solving complex problems. The process is referred to as stochastic because it is dependent upon the use of random numbers. Modeling construction productivity is suited to Monte Carlo simulation because construction productivity is based upon input parameters that will likely vary within a range of values. A Monte Carlo simulation consists of a series of iterations, or individual simulations which are used to produce a most likely representation of contractor productivity. During one simulation iteration, random values are assigned to each input parameter according to their specified
probability distribution function. The random input parameters generated during one Monte Carlo simulation iteration are placed into a CA4PRS estimate. This estimate generates a contractor productivity estimate in lane-miles for that specific iteration. By running up to as many as several thousand iterations during a Monte Carlo simulation, CA4PRS produces an overall figure for the most likely production as well as a distribution of likely productivity.
3 Probability Distribution Functions
CA4PRS probabilistic estimation requires users to assign a probability distribution function to the input parameters in both the scheduling and resource profiles. Probability distributions are statistical functions that describe the probable behavior of a variable. In a CA4PRS analysis, the variables are the input parameters. Input parameters assigned a probabilistic function will not have one precise value, but rather a range of possible or potential values. The probability distribution function describes the probability of an input parameter being assigned a particular value in this range of potential values. Probability distributions are commonly described using graphical representation. Figure 1 depicts the behavior of an unknown input parameter over a range of possible values. For this example, a common distribution called a normal distribution is depicted. Normal distributions are defined through two statistical parameters: the mean (μ) and the standard deviation (σ). The mean value is the most likely or probable value in the probability distribution being modeled. The standard deviation describes the width of the distribution and how far values are likely to be from the mean. Standard deviations can be used for assigning the probability of a value for being within a range. For instance, for a normal distribution, 68.2% of the area under the curve is within one standard deviation whereas 95.4% of the area under the curve is within two standard deviations. Other distributions will have different shapes and descriptive parameters but are used for describing the probability of an input parameter having different values within a specified range.
Figure 2 – A uniform distribution (Wikipedia contributors, 2006)
3.3 Normal Distribution
Normal distributions are one of the most frequently used forms of distribution and are commonly known as bell curves (Weisstein, 2004). A normal distribution is a distribution that is symmetric about the mean. The distribution of values around the mean is described by the standard deviation of the sample data being represented. Assigning a normal distribution to any of the CA4PRS inputs requires input of both the mean and the standard deviation for the input being modeled. Normal distributions typically arise where a large number of small effects act additively or independently upon a variable (Wikipedia contributors, 2006). In using this type of distribution, users are required to identify an appropriate standard deviation that will describe how the input parameter will vary. If the input parameter is predicted to be fairly consistent, then a smaller standard deviation should be used. At greater levels of uncertainty, the standard deviation should be increased for CA4PRS input parameters. A recommended arbitrary starting point for unknown data is to assume the value of the standard deviation will be 10-20% of the expected input parameter mean.
Figure 3 - Normal distribution (Wikipedia contributors, 2006)
3.4 Log Normal Distribution
A log-normal probability distribution is the probability distribution of a variable whose logarithm is normally distributed. Lognormal distributions arise when a random input parameter is multiplicatively influenced by a small number of independent variables (Wikipedia contributors, 2006). With this type of distribution the value of the input parameter changes logarithmically in relation to the probability function. Demolition truck loading times and the arrival rate of material trucks are activities that can be represented by this type of distribution. On the I-10 project in California, analysis of demolition truck loading and end dump truck arrival both produced log normal distributions (Lee et al., 2001).
Figure 4 - Log normal distribution (Wikipedia contributors, 2006)
Figure 6 - Beta distribution (Wikipedia contributors, 2006)
3.7 Geometric Distribution
A geometric distribution refers to a unique type of distribution that is modeled with the statistical equation: P(X=n) = (1- p ) n-1 p This equation describes the probability of achieving a success or outcome “ p ”, for a statistical event on the nth attempt. The probability of a failure on the first try would be 1- p. The probability of a failure on n-1 trials would be (1-p)n-1^. Accordingly, the probability of a success on the nth attempt would be p , leading to the distribution described by the previously depicted equation. This distribution is commonly described through a coin flip analogy. The probability of flipping heads on any trial is ½, so p = 0.5. A success P will be defined as flipping the coin with the head up. The probability of a success P on the first trial is 0.5. The probability of seeing a success on the second trial is: P = (1-0.5) (2-1)^ × 0.5. The probability of a success on the third trial would be: P= (1-0.5) (3-1)^ × 0.5. The probability for achieving a success on trials one through six are displayed in Table 1. Input parameters that display this type of behavior can be graphically modeled with the
distribution shape shown in Figure 7. None of the CA4PRS input parameters will likely be modeled by this type of distribution. Table 1 - Geometric Distribution Probability Distribution For A Coin Toss n th Trial
Probability of a Success on nth trial 1 0. 2 0. 3 0. 4 0. 5 0. 6 0.
0
1 2 3 4 5 6 Trial Number
Probability of Success On nth
Trial
Figure 7 - Graphical representation of a geometric distribution.
3.8 Truncated Normal Distribution
A truncated normal distribution is very similar to a normal distribution, but is confined between an upper and a lower limit. To use this type of distribution CA4PRS requires inputting the mean, standard deviation, maximum and minimum values for an input parameter. This type of distribution could be used to describe an input parameter such as truck arrival rates when a minimum or maximum number of truck arrivals is known.
3.9 Truncated Log Normal Distribution
A truncated log normal distributions is very similar to a log normal distribution, but is confined between an upper and a lower limit. The value of a variable will change
(5) Demolition to PCCP installation lag time The type of distribution and distribution parameters applied to the input parameters will be strongly influenced by many factors including the type of construction closure and construction sequencing. Readers should note that the data presented in the following sections is specific to one project and is not necessarily applicable to a future projects with significantly different conditions.
4.1.1 Mobilization
The documentation reviewed on the Californian reconstruction projects in Devore and Pomona does not contain any information on the distribution of mobilization time requirements. On the I-5 James to Olive Streets Pavement Rehabilitation project WSDOT construction inspection personnel collected mobilization time requirements from four closure windows. A data sample from four construction closures does not provide a large or comprehensive representation of input parameter variability and behavior. The mobilization times observed on this project are depicted in Table 2. This limited data does not depict an easily recognizable distribution. Mobilization times could be logically assumed to have either a triangular, normal, and log-normal distributions. Table 2 - Mobilization Times From The I-5 James To Olive Pavement Reconstruction Project. Construction Closure
Mobilization Time (hrs.) Stage 1 0: Stage 2 1: Stage 3 1: Stage4 1:
4.1.2 Demobilization
Minimal distribution data exists for demobilization times similar to mobilization times. Recorded demobilization times from the I-5 James to Olive Streets Pavement Rehabilitation project are presented in Table 3. Demobilization times have been calculated as the time that elapsed between the conclusion of PCC paving and the completion of temporary barrier removal. The four available demobilization times do not depict an easily recognizable distribution. Dependent on available information,
demobilization times are suggested to be modeled with triangular, normal and log-normal distributions. Table 3 - Demobilization Times From The I-5 James To Olive Pavement Reconstruction Project. Construction Closure
Demobilization Time (hrs.) Stage 1 13: Stage 2 11: Stage 3 11: Stage4 11:
4.1.3 Demolition to New Base Installation
Conclusive distribution data has not been collected on demolition to new base installation lag times. Users are recommended to apply either a triangular, normal or log-normal distributions. The lag times observed during the four closures on the I-5 James to Olive Streets Rehabilitation project are presented in Table 4 to provide program users an indication of the magnitude and variability of this input parameter. Table 4 - Demolition To New Base Installation Lag Times Observed On The I-5 James To Olive Streets Pavement Rehabilitation Project. Construction Closure
End of Demo. To Start of HMA Paving (hrs) Stage 1 4. Stage 2 0. Stage 3 3. Stage4 4.
4.1.4 New Base Installation to PCCP Installation
Conclusive distribution data has not been collected on demolition to new base installation lag times. Users are recommended to apply triangular, normal and log-normal distributions. The lag times observed during the four closures on the I-5 James to Olive Streets Rehabilitation project are presented in Table 4 to provide program users an indication of the magnitude and variability of this input parameter.
4.2.1 Demolition Hauling Trucks
4.2.1.1 Trucks Per Hour Per Team
Researchers working during construction on I-10 collected demolition truck distribution information based on 466 truck trips. Analysis of the data showed that on average, 9 demolition trucks arrived per hour per team with a 2.3 truck standard deviation. The graphical distribution of truck arrival rates shows something like normal or log-normal behavior. Based on these findings, future program users are recommended to apply normal or log-normal distributions to demolition truck arrival rates.
Figure 8 - Distribution of demolition truck arrival rates recorded during construction of the I-10 Pomona, California project.
4.2.1.2 Packing Efficiency
During construction on I-10 in Pomona, California, researchers found that demolition trucks with a 22-ton capacity (9m^3 or 2.7 slabs) carried loads that ranged from 8 to 12 metric tons (3.3m^3 to 4.9m^3 ). The observed decrease in capacity was due to the inefficient packing caused by large bulky pavement sections. Although researchers collected detailed information, no data is provided for the distribution or the distribution parameters associated with demolition truck packing efficiency. Program users are recommended to apply triangular, normal and log-normal distributions.
4.2.1.3 Number of Teams
On most construction projects the number of demolition teams will be best represented as a deterministic quantity. In order to efficiently manage costs and resources, the number of demolition teams will most likely remain constant.
4.2.1.4 Team Efficiency
CA4PRS provides users with the option of establishing a team efficiency and team efficiency distribution. On the I-10 Pomona project, team efficiency was calculated based upon loading rates and the number of trucks loaded per hour. The average loading time for a demolition truck was found to be 5.5 minutes, or 10.90 demolition trucks per hour. Because each team loaded an average of 9 trucks per hour, team efficiency was calculated as 82.5% (9/10.9). Because team efficiency calculation is based upon loading rate, the distribution of team efficiency can be related to the distribution of demolition loading rates. The average load time of 5.5 minutes was found to have a standard deviation of 0.9 minutes. Based upon the distribution of truck loading times shown in Figure 9, team efficiency can accurately be represented by a log-normal distribution.
Figure 9 - The distribution of demolition loading times observed on the I-10 project in Pomona, California.
arrival standard deviation similar in magnitude to that of the demolition truck arrival standard deviation.
4.2.2.2 Efficiency
The HMA truck ticket receipts from the I-5 James to Olive Streets Pavement Rehabilitation project contain HMA truck load information that is widely distributed. The tonnage of HMA hauled per truck load varies between fifteen to thirty-four tons. The distribution of load size using 66 truck tickets data from construction stages 1 and 4 have been used to produce Figure 11. This distribution of data points shows that the contractor utilized three different types of trucks to haul HMA loads. The three different truck sizes can be approximated by 15 ton, 27 ton and 33 ton loads. This distribution shows that a contractor used the equipment that was available and not necessarily one type of truck. In order to use the largest data sample, distribution analysis will use truck ticket information from the trucks that hauled loads of 31.9 tons or more of HMA.
5
10
15
20
25
30
35
0 10 20 30 40 50 60 Truck Loads From Sampled Truck Tickets
HMA Truck Load (tons)
Figure 11- The HMA load size distribution taken from truck tickets.
Twenty-four truck tickets for the HMA trucks that carried between 31.9 tons and 33. tons of HMA have been used to produce Figure 12. There is no obvious distribution for load size, and the differences in load size appear negligible. The difference between the
average load and the maximum load is 0.2 tons. For HMA trucks carrying 31.9 to 33. tons of HMA, truck load sizes are consistent, and by correlation, packing efficiencies should also be consistent. The tight clustering of HMA loads can be explained by the fact that trucks are probably loaded close to the legal axle weight limit permissible on Washington State roads. Because of the minimal variation, HMA packing efficiency should be assigned a deterministic distribution with a mean value of 100%.
0
5
10
15
20
25
31.9 32.1 32.3 32.5 32.7 More HMA Truck Load Size (tons)
Number of Occurrences
Figure 12 - Distribution of HMA truck load size.
4.2.3 Batch Plant Capacity
For most rapid rehabilitation projects, large stationary concrete plants will likely have a production capacity that exceeds the material handling capacity of the contractor. Equipment availability, access, space restrictions and other factors will limit how much material a contractor can place. The contractor who completed the paving work on the I- 10 rehabilitation project only utilized half the hourly capacity of a plant capable of producing 170m^3 of material per hour. The tight specifications and high costs associated with rapid construction projects also decrease production variability. Contractors will likely have backup production plants and access to sufficient material supplies to meet contract quantities. Batch plant capacity should be treated as deterministic or a normal distribution with only small variation (low standard deviation).
Table 6 - PCC Truck Arrival Rates Per Hour Based Upon Truck Ticket Information 6/28/
1:00AM - 2:00AM
2:00AM - 3:00AM
3:00AM - 4:00AM
4:00AM - 5:00AM
5:00AM - 6:00AM Trucks Per Hour 12 13 15 17 15
6/18/
7:00PM - 8:00PM
8:00PM - 9:00PM
9:00PM - 10:00PM Trucks Per Hour 10 9 9
6/25/
8:00PM - 9:00PM
9:00PM - 10:00PM
10:00PM - 11:00PM
11:00PM - 12:00AM
12:00AM - 1:00AM
1:00AM - 2:00AM Trucks Per Hour 14 14 11 12 13 13
6/28/
6:00PM - 7:00PM
7:00PM - 8:00PM
8:00PM - 9:00PM
9:00PM - 10:00AM
10:00AM - 11:00AM
11:00AM - 12:00PM
12:00PM - 1:00PM Trucks Per Hour 8 14 9 11 16 11 15
7/16/
5:00PM - 6:00PM
8:00PM - 9:00PM Trucks Per Hour 16 17
8 10 12 14 16 18 More Truck Arrivals Per Hour
Frequency of Ocurrence
Figure 14 - The distribution of hourly arrival rates for PCC delivery trucks. Data collection from two different PCC paving projects depicts lognormal distribution for PCC truck arrival behavior. Users developing estimates for future are recommended to apply lognormal distributions with arrival distributions that have a standard deviation approximately 20% of the mean value.
4.2.4.2 Packing Efficiency
CA4PRS includes a packing efficiency to address material buildup issues with Fast Setting Hydraulic Cement (FSHC). The I-10 project completed in California used a concrete pavement mix that had a four hour cure time. Due to the rapid set time, material
would tend to set and adhere to the inside of the mixing drum. As more material accumulated in the drum, less space was available for material. Because of the material buildup and expense of these materials, future construction projects are not likely to use these types of materials exclusively. For non-FSHC projects users are recommended to use a deterministic distribution and a packing efficiency of 1. For FSHC projects, a value less than one with a 10% standard deviation is advised.
4.2.5 Paver Speed
For users developing future estimates on projects that contain hand and machine paving, PCC paver speed should be represented by a deterministic rate or a probabilistic distribution with a small standard deviation. Paving machines produces the best ride and pavement quality in terms of a roughness index when they maintain a consistent speed. In effort to deliver a high quality project, most contractors will try to maintain a constant paver speed.
