Optimization has been an active area of research for several decades. As many real-world optimization problems become increasingly complex, better optimization algorithms are always needed. Unconstrained optimization problems can be formulated as a D-dimensional minimization problem as follows: Min f (x) x=[x1+x2+……..xD] where D is the number of the parameters to be optimized. subjected to: Gi(x) <=0, i=1…q Hj(x) =0, j=q+1,……m Xε [Xmin, Xmax]D, q is the number of inequality constraints and m-q is the number of equality constraints. The particle swarm optimizer (PSO) is a relatively new technique. Particle swarm optimizer (PSO), introduced by Kennedy and Eberhart in 1995, [1] emulates flocking behavior of birds to solve the optimization problems. / In this paper the concept of dynamic particle swarm optimization is introduced. The dynamic PSO is different from the existing PSO’s and some local version of PSO in terms of swarm size and topology. Experiment conducted for benchmark functions of single objective optimization problem, which shows the better performance rather the basic PSO. The paper also contains the comparative analysis for Simple PSO and Dynamic PSO which shows the better result for dynamic PSO rather than simple PSO.

This thesis intends to utilize mathematical models for testing the development of sustainable urban neighbourhoods and analyze the impact of these developments at city level using dynamic and multiobjective optimization techniques. These techniques aim to monitor and lower urban carbon emission levels, while predicting the municipality’s projected tax revenues. This study shows how multiple decision making models can operate and re- late to help analyze the implementation of a sustainable neighbourhood design in a mid-size urban area.

Slotted Waveguide Antenna Stiffened Structures (SWASS) offer a new way to integrate the antennas used by many aircraft systems in modern aircraft. Looking at the weather radars used by current aircraft and using the loading estimates of the X-47B from Northrop Grumman, the designs went through several stages in the optimization procedure. The first stage centered around accounting for the stress concentrations present at the corners of the slots. These points led to local failure around the slots prior to the buckling of the overall structure, but the development of a concentration factor curve fit accounted for these in the optimization procedure and filled in a gap in the current literature. The models are then optimized, exposing a weakness in that these stress concentrations would lead to failure well before buckling in most designs with a loaded copper insert. To avoid this and shift most of the load to the supporting material, an initial gap is implemented in the eigenvalue buckling analysis, thus allowing for the simple 1-D models to be rapidly optimized without the need for contact modelling upon the gap's closure. The waveguide designs are then analyzed to ensure that the optimization of the individual waveguides is not prioritizing the structural performance to the detriment of the electromagnetic performance. Multiple points along the optimized Pareto front are tested and showed that their electromagnetic performance was consistent across the various regions of the front, and that the desired frequency of 10 GHz used by weather radars was within the optimal operational range for the various designs. Continuing from the individual waveguides now to larger panels, high fidelity models were used to develop another curve fit that relates the buckling of a panel simply supported on all four sides to the buckling of a single constituent waveguide simply supported on both ends. This curve fit is then used to validate the larger panel's performance against anticipated flight loads, without the need to model entire panels during the optimization procedure. / Doctor of Philosophy / Modern aircraft utilize antennas for a variety of purpose, ranging from the weather radars in the nose of passenger airlines, to the communications antennas mounted on the exterior of military aircraft, and even the targeting radars used by weapons systems in modern military craft. However, these systems often require large empty spaces within the aircraft or interfere with the profile of the aircraft if mounted externally. Slotted Waveguide Antenna Stiffened Structures (SWASS) aims to eliminate these issues by integrating these antennas into the skin of the aircraft but uses the antennas themselves to help strengthen the structures, thereby eliminating the need to reroute the loads around them and making the aircraft lighter. These designs consist of a slotted metallic waveguide enclosed within supporting composite materials, which are substituted in place of the standard aircraft skin so as to fit seamlessly into the designs. Multiple issues can arise when attempting to do this, which this thesis tackles. To develop optimized, multifunctional designs the thesis balances the structural needs to integrate the designs into existing aircraft against the electromagnetic needs of the antenna systems it replaces. Gaps in the existing literature are addressed through the development of a curve fit to properly account for issues caused by the slots cut into the upper surface of the waveguides. New methods are also employed to simplify the optimization procedure. The first is reducing the load on the metallic waveguide through an initial gap by deriving a simplified model and eliminating the need for the complex models previously required. The next step is the creation of a new curve fit to relate the buckling of a single, less complex single waveguide model, to the buckling of the larger, more complex panel models. Throughout all of this, constraints and model validations are used to ensure that the designs meet their requirements, both as an antenna as well as a load bearing part of the aircraft's skin, specifically that of the X-47B.

In oil/gas and chemical industries, dynamics is one of the most essential characteristics of any process. Time-dependent response is involved in most steps of both the physical/engineering processes and the equipment performance. The conventional Quantitative Risk Assessment (QRA) is unable to address the time dependent effect in such dynamic processes. In this dissertation, a methodology of Dynamic Operational Risk Assessment (DORA) is developed for operational risk analysis in oil/gas and chemical industries. Given the assumption that the component performance state determines the value of parameters in process dynamics equations, the DORA probabilistic modeling integrates stochastic modeling and process dynamics modeling to evaluate operational risk. The stochastic system-state trajectory is modeled based on the abnormal behavior or failure of the components. For each of the possible system-state trajectories, a process dynamics evaluation is carried out to check whether process variables, e.g., level, flow rate, temperature, pressure, or chemical concentration, remain in their desirable regions. Monte Carlo simulations are performed to calculate the probability of process variable exceeding the safety boundaries. Component testing/inspection intervals and repair time are critical parameters to define the system-state configuration; and play an important role for evaluating the probability of operational failure. Sensitivity analysis is suggested to assist selecting the DORA probabilistic modeling inputs. In this study, probabilistic approach to characterize uncertainty associated with QRA is proposed to analyze data and experiment results in order to enhance the understanding of uncertainty and improve the accuracy of the risk estimation. Different scenarios on an oil/gas separation system were used to demonstrate the application of DORA method, and approaches are proposed for sensitivity and uncertainty analysis. Case study on a knockout drum in the distillation unit of a refinery process shows that the epistemic uncertainty associated with the risk estimation is reduced through Bayesian updating of the generic reliability information using plant specific real time testing or reliability data. Case study on an oil/gas separator component inspection interval optimization illustrates the cost benefit analysis in DORA framework and how DORA probabilistic modeling can be used as a tool for decision making. DORA not only provides a framework to evaluate the dynamic operational risk in oil/gas and chemical industries, but also guides the process design and optimization of the critical parameters such as component inspection intervals.

In today's competitive business environment, a firm's ability to make the correct, critical decisions can be translated into a great competitive advantage. Most of these critical real-world decisions involve the optimization not only of multiple objectives simultaneously, but also conflicting objectives, where improving one objective may degrade the performance of one or more of the other objectives. Traditional approaches for solving multiobjective optimization problems typically try to scalarize the multiple objectives into a single objective. This transforms the original multiple optimization problem formulation into a single objective optimization problem with a single solution. However, the drawbacks to these traditional approaches have motivated researchers and practitioners to seek alternative techniques that yield a set of Pareto optimal solutions rather than only a single solution. The problem becomes much more complicated in stochastic environments when the objectives take on uncertain (or "noisy") values due to random influences within the system being optimized, which is the case in real-world environments. Moreover, in stochastic environments, a solution approach should be sufficiently robust and/or capable of handling the uncertainty of the objective values. This makes the development of effective solution techniques that generate Pareto optimal solutions within these problem environments even more challenging than in their deterministic counterparts. Furthermore, many real-world problems involve complicated, "black-box" objective functions making a large number of solution evaluations computationally- and/or financially-prohibitive. This is often the case when complex computer simulation models are used to repeatedly evaluate possible solutions in search of the best solution (or set of solutions). Therefore, multiobjective optimization approaches capable of rapidly finding a diverse set of Pareto optimal solutions would be greatly beneficial. This research proposes two new multiobjective evolutionary algorithms (MOEAs), called fast Pareto genetic algorithm (FPGA) and stochastic Pareto genetic algorithm (SPGA), for optimization problems with multiple deterministic objectives and stochastic objectives, respectively. New search operators are introduced and employed to enhance the algorithms' performance in terms of converging fast to the true Pareto optimal frontier while maintaining a diverse set of nondominated solutions along the Pareto optimal front. New concepts of solution dominance are defined for better discrimination among competing solutions in stochastic environments. SPGA uses a solution ranking strategy based on these new concepts. Computational results for a suite of published test problems indicate that both FPGA and SPGA are promising approaches. The results show that both FPGA and SPGA outperform the improved nondominated sorting genetic algorithm (NSGA-II), widely-considered benchmark in the MOEA research community, in terms of fast convergence to the true Pareto optimal frontier and diversity among the solutions along the front. The results also show that FPGA and SPGA require far fewer solution evaluations than NSGA-II, which is crucial in computationally-expensive simulation modeling applications.

This study explores the applicability of including groundwater recharge and water table variation as additional objective functions in a multi-objective optimization approach to design optimal groundwater monitoring networks. The study was conducted using the Ontario Provincial Groundwater Monitoring Network wells in the Hamilton, Halton, and Credit Valley regions in southern Ontario. The Dual Entropy-Multiobjective Optimization (DEMO) model which has been demonstrated to be sufficiently robust for designing optimum hydrometric networks was used in these analyses. The importance of determining the applicability in using additional design objectives in DEMO, including groundwater recharge and groundwater table seasonal variation, is rooted in the limitations of groundwater data and the time required setting up the models. While recharge allows for the capturing of spatial variability of climate, geomorphology, and geology of the area, the groundwater table series reflect the temporal/seasonal variability. The two set of information are complementary and should provide additional information to the DEMO for optimal network design. Two sources of groundwater recharge data were examined and compared; the recharge provided by the local conservation authorities, calculated using both the Precipitation-Runoff Modeling System (PRMS) and Hydrological Simulation Program--Fortran (HSP--F), and the recharge calculated in situ using only PRMS. The entropy functions are used to identify optimal trade-offs between the maximum possible information content and the minimum shared information between each of the existing and potential monitoring wells. The additional objective functions are used here to quantify the hydrological characteristics of the vadose zone in the aquifer as well as the potential impacts of agricultural, municipal, and industrial uses of groundwater in the area, and thus provide more information for the optimization algorithm to use. Results show that including additional design objectives significantly increases the number of optimal network solutions and provides additional information for potential monitoring well locations. These results suggest that it is worthwhile to include recharge as a design objective if the data is available, and to include groundwater table variation for the design of monitoring wells for shallow groundwater system. / Thesis / Master of Applied Science (MASc)

<p> In this thesis parametric analysis for conic quadratic optimization problems is studied. In parametric analysis, which is often referred to as parametric optimization or parametric programming, a perturbation parameter is introduced into the optimization problem, which means that the coefficients in the objective function of the problem and in the right-hand-side of the constraints are perturbed. First, we describe linear, convex quadratic and second order cone optimization problems and their parametric versions. Second, the theory for finding solutions of the parametric problems is developed. We also present algorithms for solving such problems. Third, we demonstrate how to use parametric optimization techniques to solve multiobjective optimization problems and compute Pareto efficient surfaces. </p> <p> We implement our novel algorithm for hi-parametric quadratic optimization. It utilizes existing solvers to solve auxiliary problems. We present numerical results produced by our parametric optimization package on a number of practical financial and non-financial computational problems. In the latter we consider problems of drug design and beam intensity optimization for radiation therapy. </p> <p> In the financial applications part, two risk management optimization models are developed or extended. These two models are a portfolio replication framework and a credit risk optimization framework. We describe applications of multiobjective optimization to existing financial models and novel models that we have developed. We solve a number of examples of financial multiobjective optimization problems using our parametric optimization algorithms. </p> / Thesis / Doctor of Philosophy (PhD)

El objetivo es desarrollar una herramienta de apoyo a la toma de decisiones para la planificación estratégica de cadenas de suministro (CS). La tarea consiste en determinar el número, ubicación y capacidad de todos los nodos de la CS, su política de expansión, el transporte y la producción entre todos los nodos de la red. El problema se formula como un modelo de programación lineal entera mixta (MILP) que se resuelve utilizando diferentes herramientas. En primer lugar se desarrolló una estrategia de descomposición para acelerar el proceso de resolución En segundo, se utilizó el algoritmo de aproximación para resolver el problema MILP estocástico. Por último, el modelo multi-objetivo incorpora las soluciones de compromiso entre los aspectos económicos y ambientales. Todas las formulaciones se aplicaron al caso real de la industria de caña de azúcar en Argentina. El objetivo de las herramientas es ayudar a los responsables de planificación estratégica de las infraestructuras para la producción de productos químicos. / The aim of this thesis is to provide a decision-support tool for the strategic planning of supply chains (SCs). The task consists of determining the number, location and capacities of all SC facilities, their expansion policy, the transportation links that need to be established, and the production rates and flows of all materials involved in the network. The problem is formulated as a mixed-integer linear programming (MILP) model, which is solved using several mathematical programming tools. First, a decomposition strategy was developed to expedite the solving procedure. Second, the approximation algorithm was utilized to solve the stochastic version of the MILP. Finally, the multi-objective model was developed to incorporate the trade-off between economical and ecological issues. All formulations were applied to a real case based on the Argentinean sugarcane industry. The tools presented are intended to help policy-makers in the strategic planning of infrastructures for chemicals production.

This research aims to improve our understanding of multiobjective optimization, by comparing the performance of five multiobjective optimization algorithms, and by proposing a new formulation to consider input uncertainty in multiobjective optimization problems. Four deterministic multiobjective optimization algorithms and one probabilistic algorithm were compared: the Weighted Sum, the Adaptive Weighted Sum, the Normal Constraint, the Normal Boundary Intersection methods, and the Nondominated Sorting Genetic Algorithm-II (NSGA-II). The algorithms were compared using six test problems, which included a wide range of optimization problem types (bounded vs. unbounded, constrained vs. unconstrained). Performance metrics used for quantitative comparison were the total run (CPU) time, number of function evaluations, variance in solution distribution, and numbers of dominated and non-optimal solutions. Graphical representations of the resulting Pareto fronts were also presented. No single method outperformed the others for all performance metrics, and the two different classes of algorithms were effective for different types of problems. NSGA-II did not effectively solve problems involving unbounded design variables or equality constraints. On the other hand, the deterministic algorithms could not solve a problem with a non-continuous objective function. In the second phase of this research, design under uncertainty was considered in multiobjective optimization. The effects of input uncertainty on a Pareto front were quantitatively investigated by developing a multiobjective robust optimization framework. Two possible effects on a Pareto front were identified: a shift away from the Utopia point, and a shrinking of the Pareto curve. A set of Pareto fronts were obtained in which the optimum solutions have different levels of insensitivity or robustness. Four test problems were used to examine the Pareto front change. Increasing the insensitivity requirement of the objective function with regard to input variations moved the Pareto front away from the Utopia point or reduced the length of the Pareto front. These changes were quantified, and the effects of changing robustness requirements were discussed. The approach would provide designers with not only the choice of optimal solutions on a Pareto front in traditional multiobjective optimization, but also an additional choice of a suitable Pareto front according to the acceptable level of performance variation. / Thesis (Master, Mechanical and Materials Engineering) -- Queen's University, 2009-08-10 21:59:13.795

The focus of this research is to provide an efficient approach to deal with computationally expensive Multiobjective Optimization Problems (MOP’s). Typically, approximation or surrogate based techniques are adopted to reduce the computational cost. In such cases, the original expensive objective function is replaced by a cheaper mathematical model, where this model mimics the behavior/input-output (i.e. design variable – objective value) relationship. However, it is difficult to model an exact substitute of the targeted objective function. Furthermore, if this kind of approach is used in an evolutionary search, literally, the number of function evaluations does not reduce (i.e. The number of original function evaluation is replaced by the number of surrogate/approximate function evaluation). However, if a large number of individuals are considered, the surrogate model fails to offer smaller computational cost. / To tackle this problem, we have reformulated the concept of surrogate modeling in a different way, which is more suitable for the Multiobjective Evolutionary Algorithm(MOEA) paradigm. In our approach, we do not approximate the objective function; rather we model the input-output behavior of the underlying MOEA itself. The model attempts to identify the search path (in both design-variable and objective spaces) and from this trajectory the model is guaranteed to generate non-dominated solutions (especially, during the initial iterations of the underlying MOEA – with respect to the current solutions) for the next iterations of the MOEA. Therefore, the MOEA can avoid re-evaluating the dominated solutions and thus can save large amount of computational cost due to expensive function evaluations. We have designed our approximation model as a variation operator – that follows the trajectory of the fronts and can be “plugged-in” to any kind of MOEA where non-domination based selection is used. Hence it is termed– the “Pareto-Following Variation Operator (PFVO)”. This approach also provides some added advantage that we can still use the original objective function and thus the search procedure becomes robust and suitable, especially for dynamic problems. / We have integrated the model into three base-line MOEA’s: “Non-dominated Sorting Genetic Algorithm - II (NSGA-II)”, “Strength Pareto Evolutionary Algorithm - II (SPEAII)”and the recently proposed “Regularity Model Based Estimation of Distribution Algorithm (RM-MEDA)”. We have also conducted an exhaustive simulation study using several benchmark MOP’s. Detailed performance and statistical analysis reveals promising results. As an extension, we have implemented our idea for dynamic MOP’s. We have also integrated PFVO into diffusion based/cellular MOEA in a distributed/Grid environment. Most experimental results and analysis reveal that PFVO can be used as a performance enhancement tool for any kind of MOEA.