Descripción de la Propuesta de Auditoria

In this section, the impact of variability on the cost and time of each of the three structures of the highway project, and on the overall project, is assessed. In order to evaluate the impact of variability, the established deterministic total cost and total time of highway structures (earthwork, bridge and pavement) were compared to the 90th percentile of the developed lognormal models for cost variability and triangular models for time variability of highway structures. The 90th percentiles of the cost and time of variability models were estimated by simulating the cost and time of the activities and overall project, generated by the Monte Carlo simulation. The Monte Carlo simulation was run until the total of the standard deviation of structures was bounded within ±1% in 10 out of 10 sample simulations. The deterministic total cost and time (red) and the samples of Monte Carlo simulation of variability of cost and time of earthwork structure (25,100 runs), bridge structure (17,500), pavement structure (21,700) and overall (52,800) project (black) are illustrated respectively in Figures 6.17 to 6.20.

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Figure 6.17: Deterministic and variation cost and time of the earthwork structure

Figure 6.18: Deterministic and variation cost and time of the bridge structure

0 50 100 150 200 250 0 50 100 150 200 250 300 350 DAY MILLION RAND 0 50 100 150 200 250 300 350 400 450 0 50 100 150 200 DAY MILLION RAND

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Figure 6.19: Deterministic and variation cost and time of the pavement structure

Figure 6.20: Deterministic and variation cost and time of the project

The deterministic cost and time were a single value (red dot), while the results of modelling variability of cost and time were a cloud of values (black cloud dots). The 90th percentiles of the total cost and total time of variation models were selected to compare with the deterministic total cost and total time of the main structures of the highway project, because the 90th percentile is located in the upper tail of both lognormal and triangular distribution models, which covers a large part of the variability distributions; also the 90th percentile shows that there is only a 10% chance that the deterministic cost or time of structures is higher than the total cost and total time of modelled variabilities, which is assumed to be acceptable.

0 50 100 150 200 250 0 200 400 600 800 DAY MILLION RAND 0 50 100 150 200 250 300 350 400 450 0 200 400 600 800 1000 DAY MILLION RAND

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The 90th _{percentile is employed as the standard point by transportation agencies such as the South} African National Roads Agency (SANRAL) to evaluate and budget for highway projects (decision statistic point) (Reilly et al., 2004, Caltrans, 2012, WSDOT, 2018).

The 90th percentile of lognormal distribution (cost) and the 90th percentile of triangular distribution (time) were calculated with Equations 6.2 and 6.3, respectively.

𝐶𝑜𝑠𝑡 𝑃90𝑡ℎ = 1.28 × [[exp(𝜎2) − 1](exp(2µ + 𝜎2))] + exp(µ + 𝜎2) [6.2] 𝑇𝑖𝑚𝑒 𝑃90𝑡ℎ=

𝑎𝑏−𝑐(𝑎+𝑏−1.8)−0.81

(𝑐−𝑎)×(𝑐−𝑏) [6.3]

Where µ and σ are respectively the mean and standard deviation of the modelled cost variability (lognormal) and a, b and c are the minimum, mode and maximum of the modelled time variabilities (triangular) respectively.

The increases in total cost and time of the highway structures were quantified by comparing the 90th percentiles of the total cost and total time variation models with the deterministic total cost and total time of the structure as shown with Equations 6.4 and 6.5:

𝐼𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑐% = 𝐶𝑝90− 𝐶𝑑 𝐶𝑑 [6.4] 𝐼𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑇% = 𝑇𝑝90− 𝑇𝑑 𝑇𝑑 [6.5] Where CP90 is the 90th percentile of the total cost variation model, Cd is the deterministic total cost, TP90 is the 90th _{percentile of the total time variation model and T}

d is the deterministic total time.

The cost variation and time variation of the three structures of the project and overall highway were calculated and are presented in Table 6.9.

Table 6.9: Deterministic cost and time and variability distributions parameters of the three structures and the project

Earthwork Bridge Pavement Overall project Deterministic Cost 244,292,292.83 70,952,700.00 494,259,930.00 809,504,922.80 Time 215 340 217 340 Mean Cost 248,195,137.45 72,341,460.00 501,019,564.69 821,556,162.14 Time 223.67 341.67 217.67 341.67 Standard division Cost 8.80 19.59 6.62 19.59 Time 7,916,743.40 3,714,411.20 12,692,110.77 15,431,225.89 90th_percentile Cost 258,410,039.21 76,883,575.67 517,582,030.41 841,927,264.18 Time 228.76 369.48 228.53 368.64 Increased 90th_percentile Cost 5.78% 8.36% 4.72% 4.01% Time 6.40% 8.67% 5.31% 8.42% Modified variation Cost 4.12% 6.28% 3.31% 2.48% Time 2.28% 8.14% 4.99% 7.89%

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As shown in Table 6.9, the variability of cost and time (90th _{percentile) of all three structures and} the overall project were larger than the relevant deterministic cost and time (mode cost and time). The investigation of deterministic cost and time and estimated variation cost and time from simulation revealed the following:

• The deterministic cost and time were a single value (red dot), while the results of modelling variability of cost and time were a cloud of values (black cloud dots). This cloud of values revealed that the variability of cost and time were uncertain values and proved that variability was one of the main sources of uncertainty of cost and total time of the three structures and overall highway project.

• The deterministic values of all three structures and the overall project were located in the left part of the variability cloud values since the deterministic costs were closer to the most likely cost estimated by experts (mode is smaller than the mean in lognormal distribution). • The time variability models of the three structures and overall project were skewed to the

right because the most likely duration for these structures and overall project were less than the mean of modelled variability of time. Therefore, the deterministic times are located at lower parts of the variability cloud values.

For all three structures and the overall project, the 90th percentiles of the cost and time distributions were larger than the mode of cost and time variability model (cost and time deterministic were estimated based on the mode). These variations in cost and time were the main reason for creating the cloud of values in scatterplots and the magnitude of uncertainty in the variability of cost and time. For instance, the size of this variation in the time (8.42%) of the overall project was larger than the cost (4.01%) dispersion. Thus, the clouds of the overall project were scattered wider along the Y-axis (time) compared to the X-axis (cost), which means the magnitude of uncertainty in the time of the project is higher than its cost.

Similarly, the magnitude of uncertainty in the time of all three structures is higher than the uncertainty in their cost (earthwork 5.78% - 6.40%, bridge 8.36% - 8.67%, pavement 4.72% - 5.53%). Further investigation of the cost dispersion and time dispersion of the three structures disclosed that the bridge structure has the highest magnitude of uncertainty among other structures at 8.36% in cost and 8.67%, in time. This observation manifested that the bridge structure drives the cost and time variability uncertainty in a highway project.

The practice of calculating the deterministic total cost and total time solely based on the mode (most likely) input cost and time (Molenaar, 2010, and WSDOT, 2012) is controversial, because the deterministic total cost and total time estimated based on the mode are smaller than the mean of possible variability modelled, due to positive skewing of both lognormal and triangular distributions, as shown in the cumulative distribution of cost and time of the three structures and overall project. Thus, the total cost and time should be estimated based on the mean of variability ranges.

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By improving the basis of estimation of cost and time from mode (most likely) to the mean of variability, the range of uncertainty of the total cost and total time can be reduced. For instance, the magnitude of variability uncertainty in the earthwork structure could be reduced from 5.78% to 4.12% in cost and from 6.40% to 2.28% in time.