No description available.

In the last 30 to 40 years, many researchers have combined to build the knowledge base of theory and solution techniques that can be applied to the case of differential equations which include the effects of noise. This class of ``noisy'' differential equations is now known as stochastic differential equations (SDEs). Markov diffusion processes are included within the field of SDEs through the drift and diffusion components of the Itô form of an SDE. When these drift and diffusion components are moderately smooth functions, then the processes' transition probability densities satisfy the Fokker-Planck-Kolmogorov (FPK) equation -- an ordinary partial differential equation (PDE). Thus there is a mathematical inter-relationship that allows solutions of SDEs to be determined from the solution of a noise free differential equation which has been extensively studied since the 1920s. The main numerical solution technique employed to solve the FPK equation is the classical Finite Element Method (FEM). The FEM is of particular importance to engineers when used to solve FPK systems that describe noisy oscillators. The FEM is a powerful tool but is limited in that it is cumbersome when applied to multidimensional systems and can lead to large and complex matrix systems with their inherent solution and storage problems. I show in this thesis that the stochastic Taylor series (TS) based time discretisation approach to the solution of SDEs is an efficient and accurate technique that provides transition and steady state solutions to the associated FPK equation. The TS approach to the solution of SDEs has certain advantages over the classical techniques. These advantages include their ability to effectively tackle stiff systems, their simplicity of derivation and their ease of implementation and re-use. Unlike the FEM approach, which is difficult to apply in even only two dimensions, the simplicity of the TS approach is independant of the dimension of the system under investigation. Their main disadvantage, that of requiring a large number of simulations and the associated CPU requirements, is countered by their underlying structure which makes them perfectly suited for use on the now prevalent parallel or distributed processing systems. In summary, l will compare the TS solution of SDEs to the solution of the associated FPK equations using the classical FEM technique. One, two and three dimensional FPK systems that describe noisy oscillators have been chosen for the analysis. As higher dimensional FPK systems are rarely mentioned in the literature, the TS approach will be extended to essentially infinite dimensional systems through the solution of stochastic PDEs. In making these comparisons, the advantages of modern computing tools such as computer algebra systems and simulation software, when used as an adjunct to the solution of SDEs or their associated FPK equations, are demonstrated.

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 172-181).

This thesis studies a single item periodic review inventory problem with stochastic demand, random price and quotation cost. It differs from the traditional inventory model in that at the beginning of each period, a decision is made whether to pay the quotation cost to get the price information. If it is decided to request a price quote then the next decision is on how many units to order; otherwise, there will be no order. An (r, S1, S2) policy with r < S2, S1 <= S2 is proposed for the problem with two prices. It prescribes that when the inventory is less than or equal to r, the price quotation is requested; if the higher price is quoted, then order up to S1, otherwise to S2. There are two cases, r < S1 or S1 <= r. In the first case, every time the price is quoted, an order is placed. It is a single reorder point two order-up-to levels policy that can be considered as an extension of the (s, S) policy. In the second case, S1 <= r, it is possible to “request a quote but not buy” if the quoted price is not favorable when the inventory is between S1 and r. Two total cost functions are derived for the cases r < S1 <= S2 and S1 <= r < S2 respectively. Then optimization algorithms are devised based on the properties of the cost functions and tested in numerical study. The algorithms successfully find the optimal policies in all of the 135 test cases. Compared to the exhaustive search, the running time of the optimization algorithm is reduced significantly. The numerical study shows that the optimal (r, S1, S2) policy can save up to 50% by ordering up to different levels for different prices, compared to the optimal (s, S) policy. It also reveals that in some cases it is optimal to search price speculatively, that is with S1 < r, to request a quote but only place an order when the lower price is realized, when the inventory level is between S1 and r.

This thesis studies a single item periodic review inventory problem with stochastic demand, random price and quotation cost. It differs from the traditional inventory model in that at the beginning of each period, a decision is made whether to pay the quotation cost to get the price information. If it is decided to request a price quote then the next decision is on how many units to order; otherwise, there will be no order. An (r, S1, S2) policy with r < S2, S1 <= S2 is proposed for the problem with two prices. It prescribes that when the inventory is less than or equal to r, the price quotation is requested; if the higher price is quoted, then order up to S1, otherwise to S2. There are two cases, r < S1 or S1 <= r. In the first case, every time the price is quoted, an order is placed. It is a single reorder point two order-up-to levels policy that can be considered as an extension of the (s, S) policy. In the second case, S1 <= r, it is possible to “request a quote but not buy” if the quoted price is not favorable when the inventory is between S1 and r. Two total cost functions are derived for the cases r < S1 <= S2 and S1 <= r < S2 respectively. Then optimization algorithms are devised based on the properties of the cost functions and tested in numerical study. The algorithms successfully find the optimal policies in all of the 135 test cases. Compared to the exhaustive search, the running time of the optimization algorithm is reduced significantly. The numerical study shows that the optimal (r, S1, S2) policy can save up to 50% by ordering up to different levels for different prices, compared to the optimal (s, S) policy. It also reveals that in some cases it is optimal to search price speculatively, that is with S1 < r, to request a quote but only place an order when the lower price is realized, when the inventory level is between S1 and r.

published_or_final_version / Mathematics / Master / Master of Philosophy

We consider a scheduling problem where each job requires multiple classes of resources, which we refer to as the multiple resource constrained scheduling problem(MRCSP). Potential applications include team scheduling problems that arise in service industries such as consulting and operating room scheduling. We focus on two general cases of the problem. The first case considers uncertainty of processing times, due dates, and resource availabilities consumption, which we denote as the stochastic MRCSP with uncertain parameters (SMRCSP-U). The second case considers uncertainty in the number of jobs to schedule, which arises in consulting and defense contracting when companies bid on future contracts but may or may not win the bid. We call this problem the stochastic MRCSP with job bidding (SMRCSP-JB).We first provide formulations of each problem under the framework of two-stage stochastic programming with recourse. We then develop solution methodologies for both problems. For the SMRCSP-U, we develop an exact solution method based on the L-shaped method for problems with a moderate number of scenarios. Several algorithmic enhancements are added to improve efficiency. Then, we embed the L-shaped method within a sampling-based solution method for problems with a large number of scenarios. We modify a sequential sampling procedure to allowfor approximate solution of integer programs and prove desired properties. The sampling-based method is applicable to two-stage stochastic integer programs with integer first-stage variables. Finally, we compare the solution methodologies on a set of test problems.For SMRCSP-JB, we utilize the disjunctive decomposition (D2 ) algorithm for stochastic integer programs with mixed-binary subproblems. We develop several enhancements to the D2 algorithm. First, we explore the use of a cut generation problem restricted to a subspace of the variables, which yields significant computational savings. Then, we examine generating alternative disjunctive cuts based on the generalized upper bound (GUB) constraints that appear in the second-stage of the SMRCSP-JB. We establish convergence of all D2 variants and present computational results on a set of instances of SMRCSP-JB.

No description available.

This thesis is concerned with certain properties of stochastic growth models. A stochastic growth model is a model of infection spread, through a population of individuals, that incorporates an element of randomness. The models we consider are variations on the contact process, the simplest stochastic growth model with a recurrent infection. Three main examples are considered. The first example is a version of the contact process on the complete graph that incorporates dynamic monogamous partnerships. To our knowledge, this is the first rigorous study of a stochastic spatial model of infection spread that incorporates some form of social dynamics. The second example is a non-monotonic variation on the contact process, taking place on the one-dimensional lattice, in which there is a random incubation time for the infection. Some techniques exist for studying non-monotonic particle systems, specifically models of competing populations [38] [12]. However, ours is the first rigorous study of a non-monotonic stochastic spatial model of infection spread. The third example is an additive two-stage contact process, together with a general duality theory for multi-type additive growth models. The two-stage contact process is first introduced in \cite{krone}, and several open questions are posed, most of which we have answered. There are many examples of additive growth models in the literature [26] [16] [29] [49], and most include a proof of existence of a dual process, although up to this point no general duality theory existed. In each case there are three main goals. The first is to identify a phase transition with a sharp threshold or ``critical value'' of the transmission rate, or a critical surface if there are multiple parameters. The second is to characterize either the invariant measures if the population is infinite, or to characterize the metastable behaviour and the time to extinction of the disease, if the population is finite. The final goal is to determine the asymptotic behaviour of the model, in terms of the invariant measures or the metastable states. In every model considered, we identify the phase transition. In the first and third examples we show the threshold is sharp, and in the first example we calculate the critical value as a rational function of the parameters. In the second example we cannot establish sharpness due to the lack of monotonicity. However, we show there is a phase transition within a range of transmission rates that is uniformly bounded away from zero and infinity, with respect to the incubation time. For the partnership model, we show that below the critical value, the disease dies out within C log N time for some C>0, where N is the population size. Moreover we show that above the critical value, there is a unique metastable proportion of infectious individuals that persists for at least e^{\gamma N}$ time for some $\gamma>0$. For the incubation time model, we use a block construction, with a carefully chosen good event to circumvent the lack of monotonicity, in order to show the existence of a phase transition. This technique also guarantees the existence of a non-trivial invariant measure. Due to the lack of additivity, the identification of all the invariant measures is not feasible. However, we are able to show the following is true. By rescaling time so that the average incubation period is constant, we obtain a limiting process as the incubation time tends to infinity, with a sharp phase transition and a well-defined critical value. We can then show that as the incubation time approaches infinity (or zero), the location of the phase transition in the original model converges to the critical value of the limiting process (respectively, the contact process). For the two-stage contact process, we can show that there are at most two extremal invariant measures: the trivial one, and a non-trivial upper invariant measure that appears above the critical value. This is achieved using known techniques for the contact process. We can show complete convergence, from any initial configuration, to a combination of these measures that is given by the survival probability. This, and some additional results, are in response to the questions posed by Krone in his original paper \cite{krone} on the model. We then generalize these ideas to develop a theory of additive growth models. In particular, we show that any additive growth model, having any number of types and interactions, will always have a dual process that is also an additive growth model. Under the additional technical condition that the model preserves positive correlations, we can then harness existing techniques to conclude existence of at most two extremal invariant measures, as well as complete convergence. / Graduate

Thesis (Ph.D)--Architecture, Georgia Institute of Technology, 2010. / Committee Chair: Godfried Augenbroe; Committee Member: Ellis Johnson; Committee Member: Pieter De Wilde; Committee Member: Ruchi Choudhary; Committee Member: Russell Gentry. Part of the SMARTech Electronic Thesis and Dissertation Collection.