The TRIUMF Ultracold Advanced Neutron (TUCAN) Collaboration is currently developing an ultracold neutron (UCN) source and a neutron electric dipole moment (nEDM) experiment. An nEDM (dn) is predicted by the Standard Model of particle physics to be no larger than dn ~ 10-31 e·cm. Measuring a neutron electric dipole moment larger than that would require a new theory. The current upper limit on the nEDM is |dn| < 1.8 x 10-26 e·cm (90% C.L.) [C. Abel et al., Phys. Rev. Lett. 124 (2020)]. UCNs are often utilized for these experiments because they can be stored for long times when contained in appropriate bottles. Thus, performing nEDM experiments with UCNs allows for longer observation times. The TUCAN Collaboration's goal is to reach an experimental statistical sensitivity of σ(dn) ≤ 1 x 10-27 e·cm (1σ) within 400 measurement days using the TUCAN source and EDM apparatus. This thesis describes methods for optimizing various components of the UCN source, UCN transport hardware, and the planned EDM apparatus to minimize the total experimental run time required by increasing the statistical sensitivity of the measurement. Because the TUCAN source and EDM apparatus are currently in development, simulations are used to study them. This thesis describes how to simulate the UCN aspects of the TUCAN source and EDM apparatus, and how to evaluate the results. The TUCAN modular energy spectrum analysis (MESA) method was developed during this thesis work, and is used to minimize the total measurement time required by simultaneously optimizing the duration of each stage of the experiment: production, transport, storage, and detection. The net result of this research is a ~ 35% reduction in the estimated measurement time required to reach the collaboration's goal when compared to baseline assumptions.
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