We consider a stochastic functional differential equation with infinite memory driven by a fractional Brownian motion with Hurst parameter $H>1/2$. We prove an existence and uniqueness result of the solution to the stochastic differential equation. We investigate the dependence of the solution on the initial condition and the existence of finite moments of the solution. Furthermore we generalize these results to wider classes of stochastic differential equations. The stochastic integral with respect to fractional Brownian motion is defined as a pathwise Riemann-Stieltjes integral.

A modified Weibull cumulative distribution function, which accounts for the effect of fiber length on the probability of failure, was used to characterize the variation in fiber tensile strength in a SCS-6/ TIMETAL 21S material system and was implemented within the framework of the NASA code MAC/GMC. A parametric study investigating the effect of repeating unit cell architecture and fiber strength distribution on the RUC-averaged ultimate composite strength and failure was performed. Multiscale progressive failure analyses of a tensile dogbone specimen were performed using FEAMAC/ ABAQUS to assess the effect of local variations in fiber strength on the global response. The effect of the RUC architecture, fiber strength distribution, and microscale/ macroscale discretization on the global response was determined. The methodology developed in this work for accounting for statistical variations in microscale properties that feed into macroscale progressive failure analyses can readily be applied to other composite material systems.

This research considers a single-item two-echelon supply chain facing a sequence of stochastic bulky customer demand with random order inter-arrival time and random demand size. The demand process is a general renewal process and the cost functions for both parties involve the renewal function and its integral. The complexity of the general renewal function causes the computational intractability in deciding the optimal order quantities, so approximations for the renewal function and its integral are introduced to address the computational complexity. Asymptotic expansions are commonly used in the literature to approximate the renewal function and its integral when the optimal decisions are relatively large compared to the mean of the inter-renewal time. However, the optimal policies do not necessarily fall in the asymptotic region. So the use of asymptotic expansions to approximate the renewal function and its integral in the cost functions may cause significant errors in decision making. To overcome the inaccuracy of the asymptotic approximation, this research proposes a modified approximation. The proposed approximation provides closed form functions for the renewal function and its integral which could be applied to various optimization problems such as inventory planning, supply chain management, reliability and maintenance. The proposed approximations are tested with commonly used distributions and applied to an application in the literature, yielding good performance. By applying the proposed approximation method to the supply chain cost functions, this research obtains the optimal policies for the decentralized and the centralized cases. The numerical results provide insights into the cost savings realized by the centralization of the supply chain compared to the decentralized case. Furthermore, this research investigates coordination schemes for the decentralized case to improve the utilities of parties. A cost sharing mechanism in which the vendor offers the retailer a contract as a compensation of implementing vendordesired inventory policy is investigated. The sharing could be realized by bearing part of the retailer’s inventory holding cost or fixed cost. The contract is designed to minimize the vendors cost while satisfying the individual rationality of the retailer. Other forms of coordination mechanisms, such as the side payment and delayed payment, are also discussed.

Inventory control problems constitute one of the most important research problems due to their connection with real life applications. Naturally, real life is full of uncertainty so are the most of the inventory problems. Unfortunately, it is a very challenging task to manage inventories effectively especially under uncertainty. This dissertation mainly deals with single-item, periodic review, and stochastic dynamic inventory control problems particularly on replenishment cycle control rule known as the (R, S) policy. Contribution of this thesis is multiold. In each chapter a particular research question is investigated. At the end of the day, we will be showing that non-stationary (R, S) policies are indispensable not only for its cost efficiency but its effectiveness and practicality. More specifically, the non-stationary (R, S) policy provides a convenient, efficient, effective, and modular solution for non-stationary stochastic inventory control problems.

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