The focus of this dissertation is solution strategies for stochastic mixed-integer programs with special structures. Motivation for the methods comes from the relatively sparse number of algorithms for solving stochastic mixed-integer programs. Two stage models with finite support are assumed throughout. The first contribution introduces the nodal decision framework under private information restrictions. Each node in the framework has control of an optimization model which may include stochastic parameters, and the nodes must coordinate toward a single objective in which a single optimal or close-to-optimal solution is desired. However, because of competitive issues, confidentiality requirements, incompatible database issues, or other complicating factors, no global view of the system is possible. An iterative methodology called the nodal decomposition-coordination algorithm (NDC) is formally developed in which each entity in the cooperation forms its own nodal deterministic or stochastic program. Lagrangian relaxation and subgradient optimization techniques are used to facilitate negotiation between the nodal decisions in the system without any one entity gaining access to the private information from other nodes. A computational study on NDC using supply chain inventory coordination problem instances demonstrates that the new methodology can obtain good solution values without violating private information restrictions. The results also show that the stochastic solutions outperform the corresponding expected value solutions. The next contribution presents a new algorithm called scenario Fenchel decomposition (SFD) for solving two-stage stochastic mixed 0-1 integer programs with special structure based on scenario decomposition of the problem and Fenchel cutting planes. The algorithm combines progressive hedging to restore nonanticipativity of the first-stage solution, and generates Fenchel cutting planes for the LP relaxations of the subproblems to recover integer solutions. A computational study SFD using instances with multiple knapsack constraint structure is given. Multiple knapsack constrained problems are chosen due to the advantages they provide when generating Fenchel cutting planes. The computational results are promising, and show that SFD is able to find optimal solutions for some problem instances in a short amount of time, and that overall, SFD outperforms the brute force method of solving the DEP.

This dissertation studies a framework in support electric vehicle (EV) charging station expansion and management decisions. In the first part of the dissertation, we present mathematical model for designing and managing electric vehicle charging stations, considering both long-term planning decisions and short-term hourly operational decisions (e.g., number of batteries charged, discharged through Battery-to-Grid (B2G), stored, Vehicle-to-Grid (V2G), renewable, grid power usage) over a pre-specified planning horizon and under stochastic power demand. The model captures the non-linear load congestion effect that increases exponentially as the electricity consumed by plugged-in EVs approaches the capacity of the charging station and linearizes it. The study proposes a hybrid decomposition algorithm that utilizes a Sample Average Approximation and an enhanced Progressive Hedging algorithm (PHA) inside a Constraint Generation algorithmic framework to efficiently solve the proposed optimization model. A case study based on a road network of Washington, D.C. is presented to visualize and validate the modeling results. Computational experiments demonstrate the effectiveness of the proposed algorithm in solving the problem in a practical amount of time. Finding of the study include that incorporating the load congestion factor encourages the opening of large-sized charging stations, increases the number of stored batteries, and that higher congestion costs call for a decrease in the opening of new charging stations. The second part of the dissertation is dedicated to investigate the performance of a collaborative decision model to optimize electricity flow among commercial buildings, electric vehicle charging stations, and power grid under power demand uncertainty. A two-stage stochastic programming model is proposed to incorporate energy sharing and collaborative decisions among network entities with the aim of overall energy network cost minimization. We use San Francisco, California as a testing ground to visualize and validate the modeling results. Computational experiments draw managerial insights into how different key input parameters (e.g., grid power unavailability, power collaboration restriction) affect the overall energy network design and cost. Finally, a novel disruption prevention model is proposed for designing and managing EV charging stations with respect to both long-term planning and short-term operational decisions, over a pre-determined planning horizon and under a stochastic power demand. Long-term planning decisions determine the type, location, and time of established charging stations, while short-term operational decisions manage power resource utilization. A non-linear term is introduced into the model to prevent the evolution of excessive temperature on a power line under stochastic exogenous factors such as outside temperature and air velocity. Since the re- search problem is NP-hard, a Sample Average Approximation method enhanced with a Scenario Decomposition algorithm on the basis of Lagrangian Decomposition scheme is proposed to obtain a good-quality solution within a reasonable computational time. As a testing ground, the road network of Washington, D.C. is considered to visualize and validate the modeling results. The results of the analysis provide a number of managerial insights to help decision makers achieving a more reliable and cost-effective electricity supply network.

Rozsáhlá třída inženýrských optimalizačních úloh vede na modely s omezeními ve tvaru obyčejných nebo parciálních diferenciálních rovnic (ODR nebo PDR). Protože diferenciálních rovnice je možné řešit analyticky jen v nejjednodušších případech, bylo k řešení použito numerických metod založených na diskretizaci oblasti. Zvolili jsme metodu konečných prvků, která umožňuje převod omezení ve tvaru diferenciálních rovnic na omezení ve tvaru soustavy lineárních rovnic. Reálné problémy jsou často velmi rozsáhlé a přesahují dostupnou výpočetní kapacitu. Výpočetní čas lze snížit pomocí progressive hedging algoritmu (PHA), který umožňuje paralelní implementaci. PHA je efektivní scénářová dekompoziční metoda pro řešení scénářových stochastických úloh. Modifikovaný PHA byl využit pro původní přístup prostorové dekompozice. Aproximace diferenciálních rovnic v modelu problému je dosaženo pomocí diskretizace oblasti. Diskretizace je dále využita pro prostorovou dekompozici modelu. Algoritmus prostorové dekompozice se skládá z několika hlavních kroků: vyřešení problému s hrubou diskretizací, rozdělení oblasti problému do překrývajících se částí a iterační řešení pomocí PHA s jemnější diskretizací s využitím hodnot z hrubé diskretizace jako okrajových podmínek. Prostorová dekompozice byla aplikována na základní testovací problém z oboru stavebního inženýrství, který se zabývá návrhem rozměrů průřezu nosníku. Algoritmus byl implementován v softwaru GAMS. Získané výsledky jsou zhodnoceny vzhledem k výpočetní náročnosti a délce překrytí.