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Duzkale, A K (2013) Reducing uncertainty in bid preparation phase of cost estimating for structural steel, Unpublished PhD Thesis, , Catholic University of America.

Huynh, H T (2021) Game theory applications in construction project management, Unpublished PhD Thesis, , Catholic University of America.

Malcomb, A P (2022) New universal law: Application of Tracy-Widom theory for construction network schedule resilience, Unpublished PhD Thesis, , Catholic University of America.

• Type: Thesis
• Keywords: cost overrun; effectiveness; market; game theory; construction project; integration; liquidated damages; owner; subcontractor; delay analysis; probability
• ISBN/ISSN:
• URL: https://www.proquest.com/docview/2681065091
• Abstract:
  A methodology based on random matrix theory (RMT) has been proposed to investigate the underlying behavior of project network schedules. The approach relies on a devised mathematical model and three premises. The first assumption demands that the probabilistic activity durations have an identical triangular distribution with known parameters. A repetitive joint sampling of activity durations serves to create a sample data matrix Xn×p using the identified scheme for translating a project network of size p into a random matrix utilizing its dependency structure matrix. Although the joint sampling distribution was unknown, it served to draw each of the n rows of X. The second assumption is that the Tracy-Widom (TW1) distribution is the natural distribution of each row of X's sampling. Interactions between numerous parties participating in project management and construction cause a project network schedule to fall under complex systems marked by a phase transition and a tipping point. In addition, the striking similarities between the fields of applications of the TW distributions and those of project scheduling support this assumption. The last assumption is that a project network schedule with sufficient correlation in its structure, like that of complex systems, can be investigated within the framework of RMT. This assumption is justified by the interdependence structure defined by the various pairwise links between project activities. This assumption enabled the application of RMT’s universality results to project networks schedules to investigate their underlying behavior. In RMT, the appropriately scaled eigenvalues of sample covariance matrices serve as test statistics for such a study. As a result, a carefully engineered sample covariance matrix SNETn,p,α was developed, and two standardization approaches (Norm I and Norm II) for its eigenvalues were identified. Both standardization approaches relate to the universality of the TW1 limit law, which many authors have extended (e.g., Soshnikov 2002, Péché 2008) to a broad class of matrices that are not necessarily Gaussian under relaxed assumptions. Although some of these assumptions have been eased, others must still be met. Among these extra requirements, the formulation of SNETn,p,α was chosen. Its formulation necessitated the centering and scaling of the matrix X consisting of n samples of p early finish (EF) times of a project network’s activities. In addition, it included the significance level α to test the TW1 distributional assumption. The Kolmogorov-Smirnov (K-S) goodness-of-fit test with the α values of 5, 10, and 20% was found suitable for this study. 35 project networks of diverse sizes and complexity values were identified from the study's benchmark networks 2040 obtained from the Project Scheduling Problem Library (PSPLIB). Their sizes (resp. restrictiveness RT values) ranged from 30 to 120 activities (resp. 0. 18 to 0. 69). Kelly's (1961) forward and backward passes of the critical path method (CPM) determined the EF times. Using the devised methodology, the set of 100 simulations of network schedules yielded three significant findings. First, the scatterplot of 100 pairs of the normalized largest eigenvalue (l1(n)) of SNETn,p,α and the sample size n revealed a distinct and consistent pattern. The pattern is a concave upward curve that steepens to the left and flattens to the right as n increases. Surprisingly, networks of varying sizes and complexity showed the same pattern regardless of the normalization method. Using the distributional assumption on activity durations, the deviations ∆μ of the empirical means of l1 from the mean of the TW distribution (μTW1) were determined using the same 100 outputs. They enabled the graphing of scatterplots of sample size n against ∆μ. The resulting pattern highlighted the association between n and l1. Similarly, the deviations ∆var between the variances of l1 and varTW1 were calculated. The resulting pattern, also consistent across networks, helped determine an optimum sample size (nopt ) tha would maximize variance in a project network schedule's sampled durations. This sample size was found at the mean deviation curve's intersection with the horizontal axis (n-axis). One may view nopt as the required pixel count for high-quality printing. The size nopt was found to be related to the network size p but not its RT value. Moreover, an nopt value was found for all the 35 networks and included α in the expression of SNET was not necessary. Still, leaving it out resulted in higher values of nopt . Subsequently, the derived nopt was used in a series of 1000 simulations to validate the distributional assumption on activity durations. The K-S test statistics were the normalized first through fourth-largest eigenvalues l1, l2, l3, and l4 of the matrix SNET. By comparing results based on the normalization approaches, Baik et al. (1999) and Johansson (1998)—Norm II may be better suited to studying project network scheduling behavior than Johnstone (2001)—Norm I. Under Norm I, 18 of the 35 project networks validated the null hypothesis when using l1 and l2 of their matrices SNETn,p,α. Norm II supported the null hypothesis for 19 of the 21 networks evaluated when using l3(n) and l4(n) of the matrices SNET. This discovery is significant, perhaps expected, since Baik et al. (1999) introduced Norm II while studying the length of the longest increasing sequence of random permutations, which was governed by a TW limit law. The empirical and theoretical distribution plots' agreement was displayed and compared using Q-Q plots and histograms. The graphs corroborated the K-S test results that the TW1 distribution is the limiting joint sampling distribution of project network schedules. Also, the Q-Q plots showed a proper normalization of the mth largest eigenvalue should improve the K-S test performance. After the assumed limiting distribution validation for the durations of project schedules, another methodology was proposed to help design better project schedules. The intended methodology is formulated based on the previous model and assumptions. For the matrices' eigenvalues to be standardized, the methodology's assumptions limit the sample size to nopt established at a significance level α=5%. At this α level, the TW1 distribution is the natural limiting distribution for the sampling of durations of project activities. The suggested methodology relies on three rules to help choose which principal components (PCs) to keep. The simulations on four networks of various sizes yielded the following findings. First, using the scree plot rule and proportions of each PC to the total variance of the sample covariance SNET or population correlation R matrix, the study discovered a link between nopt and PC retention for any of the networks. In addition, the eigenvalues of both SNETn,p,α and R are very nearly equal. This is a significant result. Furthermore, the investigation demonstrated that Johnston's (2001) spiked covariance model might forecast project network activities' limiting durations via a PCA-based linear regression model. On scree plots, one or a few largest eigenvalues stood out from the rest. While the proportion of total variances with an 80% cutoff criterion for selection helped select the number of the rth ranked lr(n) to retain as PCs, the hypothesis testing criteria based on TW p-values did not. The TW p-value estimations' availability for testing s after the 4th largest eigenvalues was the issue. Finally, the threshold value computation for each of the four networks indicated strong evidence of a phase transition in project network schedules transitioning from stable to unstable. This discovery is critical because it may help practitioners determine when a construction project's schedule may become problematic. Since the empirical study involved only four networks, more study is needed on PCA-based models and locating phase transitions in project network schedules. In conclusion, while the uncovered universal pattern may not be suited for manual applications, it can be added as an add-on to project netwo k scheduling applications. Doing so would aid in simulating the necessary network schedules and determining the optimal sample size corresponding to the tipping point associated with a project network schedule. At that point, a project schedule may transition from a strong-coupling phase with activities in concert to a weak-coupling phase with independent activities. In addition, because the optimal sample size corresponds to the maximum variance in the project activity durations, it may determine the limiting duration of each activity and total project duration, which, if exceeded, may result in a project schedule instability with unrecoverable delays. The proposed PCA-based linear regression model, based on Johnstone's (2001) spiked covariance model, is intended to forecast project limiting durations. These durations may aid practitioners in predicting project schedules and costs that are resilient. Finally, the significant discoveries of this pioneering study have resulted in proposals for contributions to the body of knowledge and recommendations for future research.

Schied, E P, Jr. (2012) Geometric information scheduling to identify and resolve spatial conflicts and increase efficiency of space use on construction projects, Unpublished PhD Thesis, , Catholic University of America.

Su, Y (2017) Unified quantitative modeling for integrated multi-objective project management with singularity functions, Unpublished PhD Thesis, , Catholic University of America.

Thompson, R C, Jr. (2012) Risk measurement, allocation, and pricing in network schedule systems, Unpublished PhD Thesis, , Catholic University of America.